IN  MEMORIAM 
FLORIAN  CAJORI 


AN  INDUCTIVE  MANUAL 


OF 


The  Straight  Line 


AND 


The  Circle 


WITH   MANY   EXERCISES 


BY 

WILLIAM   J.  MEYERS, 

PROFESSOR  OF  MATHEMATICS, 
THE  STATE  AGRICULTURAL  COLLEGE  OF  COLORADO. 


"  The  better  method  ....to  put  it  to  the  test  of 
Experience:'    Preface  to  Novum  Organum. 

"The  Syllogism  consists  of  Propositions, 
Propositions  of  words,  tvords  are  the  signs  of 
Notions.  If  therefore  the  tiotions  {whic/iform 
the  basis  of  the  whole)  be  confused  and  care- 
lessly abstracted  from  things,  there  is  no  solid- 
ity in  the  superstructure.  Our  only  hope,  then, 
is  in  genuine  INDUCTIONS 

Novum  Organum,  aphorism  XIV 


FORT  COLLINS,  COLORADO. 

WILLIAM  J.  MEYERS,  PUBLISHER. 

1896. 


Copyright,  1894. 
BY  WILLIAM  J.  MEYERS. 


ALL  RIGHTS  RESERVED. 


A.  M.  MEYERS,  PRINTER, 
CALEDONIA,  MICH. 


mf 


/ 


2- 


PREFACE, 


This  manual  is  published  to  supply  a  need  which  its 
author  has  long  felt,  in  the  instruction  of  his  own  class- 
es, for  some  other  mode  of  teaching  geometry  than  the 
purely  deductive,  which  is  almost  exclusively  followed 
in  most  of  our  text-books  of  to-day.  That  method 
seems  in  many  cases  to  fail  to  give  clear  ideas  to  the 
young  beginner  in  geometry  and  to  involve  the  whole 
subject  in  a  haze  which  it  takes  a  considerable  time  to 
clear  away.  In  too  many  cases,  because  he  fails  to 
comprehend  the  reasoning  employed,  he  becomes  dis- 
couraged ;  and,  putting  aside  all  confidence  in  his  own 
powers  of  thought,  he  attempts  to  convince  himself  that 
such  and  such  things  must  be  true  because  ''the  book 
says  so," — a  basis  for  opinion  which  is,  it  must  be  con- 
fessed, even  more  unsatisfactory  in  geometry  than  it  is 
in  other  things  with  which  we  concern  ourselves. 
With  such  a  student  in  such  circumstances,  the  results 
obtained  through  attempting  to  follow  the  deductive 
method  of  teaching  geometry  are  too  apt  to  be  a  befog- 
ging of  any  ideas  he  may  previously  have  had,  and  the 
direct  discouragement  of   his  powers  of  imagination. 


iv  PREFACE. 

invention,  and  judgment. 

The  course  of  work  outlined  in  the  following  pages  is 
designed  to  be  actually  and  accurately  worked  out  by 
the  student,  using  the  draughtsman's  instruments  and 
the  draughtsman's  methods  as  far  as  practicable ;  and 
each  exercise  is  to  be  worked  out  a  sufficient  number  of 
times  and  under  sufficiently  varied  conditions  for  the 
student  to  know  why  he  comes  to  such  and  such  con- 
clusions and  not  merely  to  succeed  in  guessing  what 
his  instructor  wishes  to  be  told.  Geometry  is  one  of 
the  first  sciences  in  the  historical  order  of  development, 
and  its  phenomena  are  among  the  simplest  of  all  those 
that  demand  man's  consideration.  It  offers,  then,  one 
of  the  best  means  of  training  a  student  to  exact  think- 
ing and  to  scientific  investigation.  It  is  believed  that 
with  young  students  and  with  those  whose  reasoning 
powers  have  not  received  a  practical  training  of  a  con- 
siderable extent,  a  faithful  following-out  of  an  inductive 
investigation,  such  as  it  has  been  attempted  to  indicate 
in  the  following  pages,  will  yield  as  extensive  and  exact 
a  knowledge  as  will  the  deductive  method  in  the  same 
length  of  time,  and,  besides  that,  a  much  greater  readi- 
ness in  the  application  of  the  knowledge  obtained  and  a 
much  more  thorough  training  of  the  invaluable  powers 
before-mentioned, — imagination,  invention,  and  judg- 
ment. 

It  seems  hardly  necessary  to  say  that  the  book  is  not 
particularly  designed  for  the  use  of  students  who  have 


PREFACE.  V 

not  the  benefit  of  an  instructor  (although  many  such 
will  be  able  to  use  it  to  advantage,)  and  so  is  much 
briefer  than  it  might  otherwise  have  been,  and  shows 
what  may  seem  to  some  an  alarming  paucity  of  illustra- 
tions in  connection  with  the  definitions.  Diagrams  are 
also  purposely  few,  because  it  is  believed  that  in  most 
cases  the  student  will  be  able  to  supply  his  own  dia- 
grams. Sundry  new  words  have  been  introduced  where 
there  seemed  to  be  need  of  them.  It  is  presumed  that 
their  convenience  will  offer  sufficient  justification  for 
their  use,  either  in  the  forms  here  given  or  with  such 
modifications  as  experience  may  suggest.  The  word 
"sect"  is  due  to  Prof.  Halsted  of  the  University  of  Texas. 
The  manual  herewith  offered  being  somewhat  of  an 
experiment,  criticisms  and  suggestions  are  invited  from 
those  instructors  who  may  have  occasion  to  examine 
the  work,  and  especially  from  those  who  use  it  with 
their  classes. 

William  J.  Meyers. 

Department  of  Mathematics, 

The  State  Agricultural  College, 
Fort  Collins,  Colo. 


SYMBOLS  AND  ABBREVIATIONS. 


+    Plus. 

—    Minus. 

X     Into. 

^     Divided  by. 

=      Equals. 

IT    Equivales. 

^    Approaches  as  a  limit. 

>     Is  larger  than. 

<     Is  smaller  than. 

Z     Angle. 

Zs  Angles. 

L     Right  angle. 

L  s   Right  angles. 

J_     Perpendicular. 

J_s   Perpendiculars. 

II      Parallel. 

lis     Parallels. 

A     Trigon. 

As   Trigons. 

OJ    Rhomboid. 

CJi   Rhomboids. 


□  Rectangle. 
□s   Rectangles. 

□  Square. 
^  Arc. 

O     Circumference. 
O     Circle. 
+      Given  point. 
0     Required  point. 

Given  line. 

— Hidden  line. 

Auxiliary  line. 

Required  line. 

B    Alternate. 
IT     Inner. 
©     Outer, 
adj.     Adjacent, 
sup.    Supplementary, 
cmp.  Complementary, 
exp.   Explementary. 
crsp.  Corresponding. 


I 


TABLE  OF  CONTENTS. 


PAGE 

Preface iii 

Instructions  to  the  Student xiii 

Cuts  of  Instruments xvii 

Symbols  and  Abbreviations xviii 


INTRODUCTORY  CHAPTER. 
DEFINITIONS,  CONCEPTS,  AUXILIARIES,  ETC. 

ARTICLE 

I  Appreciability i 

2-4  Equality i 

5-7  Equivalence 2 

8-9  Non-equivalence ,. . .  2 

10  Constants 2 

I I  Variables 2 

12  Limits • 3 

13-19  Quantity,  Measurements,  etc 3 

17-19  Commensurables  and  Incommensurables 5 

20-24  Ratio 6 

25-29  Proportion 7 

30  Place 7 

31  Space 8 

32  Extent 8 

33  Distance 8 

34-35  Motion  and  Path 8 

36-41  Solids,  Points, "Lines,  and  Surfaces 8 

42  Contiguity 9 

.13-44  Consecutive  and  Separate  Points. . , 10 


viii  CONTENTS. 

ARTICLE  PAGE 

45-46      Size  and  Magnitude 10 

47  Geometrical  Concepts 10 

48  Geometry 11 

49  Magnitudes  and  Figures 11 

50-51  Modes  of  representing  and  naming  concepts,  auxil- 
iaries, etc 13 

52           The  instruments  used  in  Simple  Geometry 13 


BOOK  L— RECTILINEAR  FIGURES  IN  A 
SINGLE  PLANE. 


CHAPTER  I. 

LINEAR  RELATIONS. 

53  The  Straight  Line 15 

54-55      Distance  between  Two  Points 16 

56  Ruler,  etc.    Ruled  Lines 16 

58-60      Sects  and  Semi-sects 17 

61  Broken  Line,  Chain,  Links 17 

62  CoUinear  Points 17 

63-64      Intersection.    Concurrent  Lines 17 

65  and  67  Planes  and  Plane  Figures 18 

66  Trace •. 19 

68  Geometric  Drawings 19 

69  Test  for  Straightness  of  Ruler 21 

70-73      Addition  of  Sects 22 

74  Transference  of  Sects 23 

75-77      Ratio  between  Commensurable  Sects 23 

78  The  Division  of  a  Sect  into  equal  parts  by  Trial 26 

79, 80, 81  Curve,  Arc,  Chord 26 

82-87      Circle,  Center,  Radius,  Diameter,  etc 27 

88-89      Locus 28 

90-108    Angles 29 

104         Degrees,  Minutes  and  Seconds 31 

109- 1 10  Perpendiculars ...  32 


CONTENTS.  ix 


^^H  ARTICLE  PAGE 

^^K       III          To  draw  a  Perpendicular 32 

^B       112         Obliques 33 

^H       113          Orthogonal  Projection 33 

^V       1 14-1 19  Trigons 33 

120  Obverse  and  Reverse 35 

121  Equal,  Opposite,  and  Unequal  plane  figures 35 

122  To  construct  a  trigon  whose  sides  are  equal  to  giv- 

en sects 35 

123-125  Medians,  Bisectors,  and  Altitudes 37 

126  Distance  of  a  point  from  a  line 39 

127  Draughtsman's  Triangles 40 

128-131  Symmetry 42 

1 32-1 34  Revolution 43 

1 36  Angle  at  center  of  arc 45 

137  Chord  of  angle 45 

138  Traverser  and  Traversee 46 

139-140  Parallels 47 

141  To  draw  through  a  given  point  a  line  which  shall 

be  concurrent  with  two  others  whose  point  of 

concurrence  is  off  the  sheet 50 

142  T  square  and  triangles  in  drawing  parallels 51 

143-149  Polygons 52 

150         Regular  Polygons 53 

151-153  Chains  and  Regular  Chains 53 

154-158  Circumscribed  and  Jnscribed  Rectilinear  Figures. . .  54 

159-162  Apothem,  Radius  and  Centric  Angle 56 

163-168  Tetragons 57 

169-176  Similar  Figures 60 

177         Homothesy 63 

178-181  The  Solution  of  Problems,  followed  by  a  collection 

of  problems 64 


CHAPTER  II. 

AREAL   RELATIONS. 

182-183  Areas  and  their  Measurement . .  jj 

184-185  Dimensions 77 

186  Products  and  Quotients  of  Lines,  Surfaces,  etc 78 

187  Dimensions  of  Trigons  and  of  certain  Tetragons.  . .  78 


X  CONTENTS. 

ARTICLE  PAGE 

i88  Rectangle  upon  two  given  sects 78 

189         Areas  of  Trapeziums  and  of  Polygons  of  more  than 

four  sides 81 

igo          The  Reduction  of  a  Polygon  to  an  Equivalent  Poly- 
gon of  a  less  number  of  sides 81 

191  The  relation  existing  between  the  Hypotenuse  and 

the  Two  Legs  of  a  Right  Trigon 83 

192  The  Altitude  upon  any  Side  of  a  Trigon,  and  the 

Area 85 

193  Square  Equivalent  to  a  given  Rectangle 87 

194  Mean  Proportional  between  two  Sects 87 

195  Areal  Ratio 88 


BOOK  IL— CURVILINEAR  FIGURES  IN  A 
SINGLE  PLANE. 


CHAPTER  I. 
CURVES  IN  GENERAL. 

197         Graphic  and  Mathematical  Curves 91 

199  Secant 92 

200  Tangent 92 

202  Curves  Tangent  to  Each  Other 93 

203  Curves  intersecting 93 

204  Angle  between  Curves. 94 

205  Location  of  point  of  Tangency 94 

206  Graphic  Tangent 95 

207  Inscription  and  Circumscription 96 

208-210  Rectification ^. 96 

21 1-212  Quadrature 98 

213         Similar  Curves 99 


CONTENTS  xi 

ARTICLE  PAGE 

CHAPTER  II. 
THE  CIRCLE. 

214-220  Circle,  Circumference,  Segment,  Sector,  etc 100 

221  Centric  Angle  of  Arc 102 

222  Addition  of  Arcs 102 

223  Arcs  complementary,  supplementary,  etc 102 

224  Degrees,  Minutes,  etc.,  of  arc 102 

225-226  Inscribed  Angles 103 

27         Inscription  of  mixtilinear  figures 112 


INSTRUCTIONS 

TO  STUDENTS  USING  THIS  BOOK. 


The  exercises  called  for  in  this  book  are  to  be  com- 
pletely and  accurately  worked  out,  and  every  conclu- 
sion to  which  the  student  may  come  should  be  amply 
tested  in  experience  before  being  finally  settled  upon. 
Such  drawings  as  are  indicated  by  the  instructor  should 
be  inked  in  after  approval  by  him.  These  when  bound 
together  will  form  an  appropriate  companion  to  the  stu- 
dent's note-book  in  geometry,  which  should  contain 
statements  of  his  conclusions  upon  the  various  subjects 
investigated,  and  discussions  of  the  various  problems 
proposed,  numbered  in  correspondence  with  the  num- 
bers of  the  questions  and  exercises  in  the  text-book. 
These  statements  in  the  note-book  should  be  complete, 
so  as  to  be  intelligible  without  reference  to  the  text- 
boqk,  and  in  making  them  the  student  should,  as  far  as 
he  is  able,  indicate  the  connections  existing  among  the 
various  relations  which  he  discovers.  He  will  soon  find 
that  from  the  relations  he  first  discovers  he  can  in  many 
cases  predict  what  further  relations  he  will  discover. 
This  he  should  accustom  himself  to  do ;  but  he  should  in 


xiv  INSTRUCTIONS. 

all  cases,  or  at  least  in  all  cases  except  those  in  which 
his  instructor  may  deem  it  unnecessary,  submit  his  pre- 
diction to  the  test  of  experiment,  and  his  experiments 
should  be  extensive  enough  to  cover  fairly  well  all  the 
conditions  coming  within  the  range  of  his  statements. 
By  so  doing  he  will  discover  wherein  his  statements  are 
too  broad  and  wherein  they  are  unnecessarily  narrow, 
and  will  gain  quickness  and  accuracy  in  conceiving  and 
mentally  reviewing  all  the  aspects  under  which  any 
relation  may  be  viewed,  and  will  gain  also  that  valuable 
practical  judgment  which  depends  so  largely  on  experi- 
ence. 

Concerning  Drawing  Instruments. — The  student 
will  not  need  a  large  selection  of  drawing  instruments 
but  those  he  has  should  be  good.  He  will  need  at  least 
one  3^"  compasses,  with  fixed  needle-point,  and  pen- 
and  pencil  attachments,  both  legs  jointed;  one  3^" 
dividers;  one  5"  ruling-pen;  two  triangles,  one  45°-45'- 
90°  and  the  other  30' -60° -90°,  each  having  a  hypote- 
nuse about  seven  inches  long;  and  a  T  square  the 
length  of  whose  blade  is  more  than  that  of  the  diagonal 
of  the  sheet  of  paper  on  which  his  drawings  are  to  be 
made.  Besides  these  he  will  need  a  6H  pencil,  an  eras- 
er, a  bottle  of  drawing-ink  (Higgins's  American  is  best,) 
a  finely-divided  scale  six  inches  or  more  in  length, 
a  supply  of  thumb-tacks,  etc.  His  compasses  and 
dividers  should  not  be  below  the  grade  of  the  * 'Arrow* 


They  should  be  of  higher  grade  if  to  be  used  for  regular  draughting. 


INSTRUCTIONS.  xv 

German*' brand  of  the  Keuffel  and  Esser  Co.,  and  his 
ruling-pen  should  be  of  the  highest  grade ;  a  poor  ruling, 
pen  is  an  abomination.  In  making  his  purchases  the 
student  should  bear  in  mind  that  poor  drawing  instru- 
ments are  an  incessant  plague  to  the  user  and  like  cats 
have  nine  lives.  The  best  economy  is  to  purchase  high 
grade  instruments  and  get  along  with  few  if  need  be. 
Triangles  should  be  of  vulcanite  or  some  material 
more  durable  than  wood,  since  that  soon  warps  and 
twists  out  of  shape,  especially  in  a  dry  climate.  The 
T  square  is  also  preferably  of  some  material  more 
durable  than  wood,  although  a  wooden  T  square  is 
much  less  objectionable  than  are  wooden  triangles. 
Very  satisfactory  drawing  pencils  are  Dixon's,  Faber's, 
Hardtmuth's,  or  Guttknecht's.  The  student's  drawing- 
board  may  be  made  of  some  soft  wood  into  which  the 
thumb-tacks  may  easily  be  pressed ;  but  if  so  made  it 
must  be  frequently  examined  to  see  that  its  standard 
edge  remains  straight.  A  piece  of  well-seasoned  half- 
inch  poplar  or  whitewood  *  will  serve  fairly  well ;  an 
edge  running  across  the  fiber  of  the  wood  should  be 
"trued  up"  and  used  as  the  edge  along  which  to  slide 
the  T  square  head . 

A  t:ut  of  a  collection  of  draughtsman's  instruments  is 
given  herewith.  It  is  taken  (by  permission)  from  the 
catalogue  of  the  Keuffel  and  Esser  Co.,  42  Ann  St., 
New  York  City,  one  of  the  foremost  firms  in  the  United 


*  California  redwood  boards,  one  inch  in  tliickness,  are  found  to  be  very  satis- 
factory in  Colorado. 


xvi  INSTRUCTIONS. 

States  dealing  in  and  manufacturing  such  instruments. 
Their  higher  grade  instruments  are  very  satisfactory, 
and  the  student  who  is  not  sure  of  getting  their  instru- 
ments through  his  stationer,  should  send  directly  to 
them,  or  buy  others  only  upon  the  advice  of  an  experi- 
enced draughtsman. 


XVll 


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DEFINITIONS,  CONCEPTS,  AUXILIARIES. 


INTRODUCTORY  CHAPTER.  * 


I. — Anything  is  appreciable  which  is  capable  of 
recognition. 

2 — One  of  two  things  is  said  to  be  equal  to  the 
other  when  either  may  be  put  for  the  other  and,  when 
so  put,  will  produce  all  the  effects  of  that  other,  or 
effects  which  are  not  appreciably  different  from  those  of 
that  other. 

3. — It  is  evident  that  if  each  single  thing  in  a  group 
is  equal  to  a  certain  definite  thing  outside  the  group, 
any  single  thing  in  the  group  is  equal  to  any  other 
single  thing  in  it. 

4. — The  sign  of  equality  is  =.  When  placed  between 
two  expressions,  it  is  read  *Ms  (or  are)  equal  to, ''or, 
more  briefly,  "equals  (or  equal)."  a-b  means  that 
the  thing  (not  necessarily  a  number)  represented  by  a 
equals  that  represented  by  b, 

*  This  chapter  is  here  inserted  because  it  belongs  here  logically  and  not  peda- 
gog^ically.  The  instructor  will  select  such  portions  of  it  as  he  needs  to  use  for 
introduction  and  leave  the  rest  for  later  discussion  or  for  reference  as  occasion 
mav  arise- 


2  INTRODUCTORY  CHAPTER. 

5. — One  of  two  things  is  said  to  be  equivalent  to 
the  other  when  the  two  can  be  conceived  to  be  divided 
into  parts  such  that  for  every  part  in  one  there  is  an 
equal  part  in  the  other,  no  part  of  either  being  taken  to 
correspond  to  more  than  one  part  of  the  other. 

6. — It  is  evident  that  if  each  of  the  things  in  a  certain 
group  is  equivalent  to  a  certain  thing  outside  the  group, 
any  one  thing  in  the  group  is  equivalent  to  any  other 
one  thing  in  it. 

7. — The  sign  of  equivalency  is  z: .  When  placed 
between  two  expressions,  it  is  read  ''is  (or  are)  equiva- 
lent to,"  or,  more  briefly,  ''equivales  (or  equivale. )" 
a^ib  means  that  the  thing  (not  necessarily  a  number) 
represented  by  a  equivales  that  represented  by  b. 

8. — The  first  of  two  things  is  said  to  be  larger  than 
the  second  when  the  first  can  conceivably  be  divided 
into  two  parts  one  of  which  equivales  the  second  thing. 
Saying  that  the  second  of  two  things  is  smaller  than 
the  first  means  the  same  as  saying  that  the  first  is  larg- 
er than  the  second. 

9. — The  signs  of  non-equivalence  are  >,  read  '*is  (or 
are)  larger  than,"— and  <,  read  *'is  (or  are)  smaller 
than." 

10. — Anything  which  continually  maintains  all  its 
properties  unchanged  is  called  a  constant. 

II. — Anything  some  or  all  of  whose  properties  change 
from  time  to  time  is  called  a  variable  with  respect  to 


INTRODUCTORY  CHAPTER.  3 

those  properties  which  change.  One  variable  is  said  to 
be  dependent  on  another  when  some  or  all  of  its  prop- 
erties are  restricted  by  those  of  that  other.  Thus,  the 
variable  weight  of  a  growing  plant  is  dependent  upon 
the  age  of  the  plant,  and  the  variable  amounts  and 
intensities  of  heat,  moisture,  light,  etc.,  that  it  receives. 
Any  variable  dependent  on  another  is  called  a  function 
of  that  other,  hi  any  collection  of  variables,  the  one 
whose  law  of  change  does  not  depend  on  that  of  any 
of  the  others  is  called  the  independent  variable. 

12. — Any  constant  to  which  a  variable  may  become 
as  nearly  equal  as  we  please  without,  however,  becom- 
ing exactly  equal  to  it  under  the  conditions  imposed,  is 
called  a  limit  of  that  variable.  Thus,  numerically,  2  is 
a  limit  of  the  variable  sum,  i  -\-  %  -{-  %  -\-  Yq -{■  etc.,  as 
we  take  more  and  more  terms,  for  we  can  not  take  a 
sufficiently  great  number  of  terms  to  make  the  sum 
exactly  2,  but  we  may  take  a  sufficiently  great  number 
to  make  the  sum  as  nearly  equal  to  2  as  we  please. 

13. — Quantity,  or  amount,  of  anything  is  that  which 
tells  how  much  with  respect  to  that  thing.  It  is  evi- 
dent that  quantity  is  purely  relative;  i.  e.,  the  quanti- 
ty of  anything  can  be  told  only  by  the  relation  between 
that  thing  and  some  other  thing  of  the  same  kind. 

14. — This  other  thing  which  is  used  as  standard  and 
in  terms  of  which  the  quantities  of  all  things  of  the 
same  kind  are  (or  may  be)  expressed  is  called  the  unit. 


4  INTRODUCTORY  CHAPTER. 

15. — Finding  the  quantity  of  anything  is  called  the 
measurement  of  that  thing.  The  thing  measured  is 
called  the  metrand.  The  unit  is  frequently  called  the 
measure.  Measurement  is  evidently  performed  essen- 
tially by  conceiving  the  metrand  to  be  divided,  or  cut 
up,  into  as  many  parts  as  possible,  all  of  which,  or  all 
but  one  of  which,  shall  each  equivale  the  unit, — and 
then  counting  the  parts  so  obtained.  Should  the  parts 
so  obtained  each  equivale  the  unit,  the  metrand  is  said 
to  be  a  multiple  of  the  unit,  and  the  unit  is  said  to  be 
a  sub=multiple,  or  an  exact  measure,  of  the  metrand. 
Should  one  of  the  parts  so  obtained  be  smaller  than  the 
unit,  some  sub-multiple  of  the  unit  is  to  be  taken  and 
used  as  an  auxiliary  unit  for  the  measurement  of  the 
part  aforesaid.  Should  this  part  equivale  a  sub-multiple 
of  the  unit,  or  should  it  and  the  unit  have  a  common 
sub-multiple,  the  metrand  is  said  to  be  commensura= 
ble  (2.  ^.,  measurable  together)  with  the  unit  used; 
otherwise  it  is  incommensurable  with  (or  in  terms  of) 
the  unit  used.  The  quantity  of  any  commensurable 
thing  is  evidently  the  name  of  the  unit  preceded  by  the 
number  telling  into  how  many  parts  equivalent  to  the 
unit  and  into  how  many  parts  equivalent  to  a  designated 
sub-multiple  of  the  unit  the  metrand  is  capable  of  being 
divided. 

16. — The  numerical  portion  of  the  quantity  of  any- 
thing is  the  enumerator  of  that  thing  relative  to  the 
unit  used. 


INTRODUCTORY  CHAPTER.  5 

17. — Two  things  are  said  to  be  incommensurable 

when  each  is  incommensurable  in  terms  of  the  other  as 
unit.  The  quantity  of  any  incommensurable  thing  can 
not  be  exactly  expressed.  We  can  approximate,  how- 
ever, as  closely  as  we  please  to  the  quantity  of  any 
incommensurable  thing ;  for,  it  will  be  noticed,  the 
remainder  or  unmeasured  portion  of  the  metrand  may 
always  be  made  less  than  the  measure  used.  Since  we 
may  take  as  small  a  sub-multiple  of  the  unit  as  we 
please  for  the  auxiliary  measure,  this  remainder  may 
evidently  be  made  as  small  as  we  please. 

18. — The  approximate  quantity,  or  amount,  of 
any  incommensurable  thing  is  the  quantity  of  that 
thing  which  is  most  nearly  equivalent  to  the  given 
incommensurable  and  is  yet  commensurable  in  terms  of 
the  smallest  sub-multiple  (of  the  unit)  which  it  is 
desired  to  use.  It  will  be  noticed  that  the  error  in  the 
approximate  quantity  of  any  incommensurable  thing  is 
never  so  large  as  half  of  the  smallest  auxiliary  measure 
used. 

19. — The  real  quantity  of  an  incommensura- 
ble thing  is  the  limit  of  the  approximate  quantity  as  the 
auxiliary  measure  used  is  taken  smaller  and  smaller 
indefinitely.     The    enumerator   of  an    incommen= 

surable  is  the  limit  of  the  enumerator  of  the  approx- 
imate commensurable,  and  evidently  can  not  be 
expressed  in  the  ordinary  arithmetic  symbols  of  num- 
ber.    Such  an  enumerator  is  called  an  incommensurable 


6  INTRODUCTORY  CHAPTER. 

number. 

We  become  aware  of  incommensurable  things  only 
through  theory.  In  practice  they  are  dealt  with  by 
means  of  their  approximate  quantities. 

20.— The  ratio  j  ^^^^^^^  \  the  first  of  two  things 
1  and  I  *^^  second  is  the  enumerator  of  the  first  when 
the  second  is  taken  as  unit.  Ratio  can  exist  between 
two  things,  evidently,  only  when  they  are  of  the  same 
kind.  By  the  ratio  between  two  numbers  is  meant  the 
ratio  between  two  things  whose  enumerators  these 
numbers  are. 

21. — If  the  two  things  whose  ratio  is  to  be  considered 
are  incommensurable  with  respect  to  each  other,  the 
ratio  is  said  to  be  an  incommensurable    ratio.     The 

approximate  enumerator  of  the  first  thing  in  terms  of 
the  second  as  unit,  when  the  two  are  incommensurable, 
is  called  the  approximate  ratio  between  them.  Evident- 
ly in  such  a  case  the  real  ratio  is  the  limit  which  the 
approximate  ratio  approaches. 

22. — The  first  of  the  two  things  concerned  in  any 
ratio  is  called  the  antecedent;  the  second  is  called  the 
consequent. 

23. — The  ratio  between  a  and  d  is  indicated  by  a  \by 
or  by  T,  or  by  a/ b, 

24. — If  the  antecedent  equivales  the  consequent,  the 
ratio  between  them  is  called  unity  and  is  said  to  be  a 


INTRODUCTORY  CHAPTER.  7 

ratio  of  equality.  If  the  antecedent  be  larger  than  the 
consequent,  the  ratio  is  a  ratio  of  larger  inequality; 
if  smaller,  the  ratio  is  one  of  smaller  inequality. 

25. — A  proportion  is  an  equality  of  ratios.  Four 
things  are  said  to  be  in  proportion  when  the  ratio  of  the 
first  to  the  second  equals  the  ratio  of  the  third  to  the 
fourth. 

26. — The  first  and  fourth  of  four  things  in  proportion 
are  called  the  extremes  of  the  proportion ;  the  other 
two  are  the  means. 

27. — Three  or  more  things  are  said  to  be  in  continued 
proportion  when  the  ratio  of  the  first  to  the  second 
equals  the  ratio  of  the  second  to  the  third,  this  ratio 
equals  the  ratio  of  the  third  to  the  fourth,  and  so  on.  If 
three  things  are  in  continued  proportion,  the  second  is 
called  a  mean  proportional  between  the  first  and  the 
third. 

28. — Two  things  are  proportional  to  two  others 
when  the  ratio  between  the  first  two  equals  that 
between  the  other  two  taken  in  the  same  order. 

29. — If  a,  b,  c,  and  d  are  in  proportion,  it  is  indicated 
thus;  a  \  b  \\  c  :  d,  read  ''a  is  to  /^  as  ^  is  to  ^/'  or 

^  =  A  ox  a/b  ^  eld,  read  "the  ratio  o\  a  \o  b  equals 

o        «, 

the  ratio  of  c  to  d''  or,  more  briefly,  ''a  to  b  equals  c  to 

flf"  or  ''a  over  b  equals  c  over  d.'' 

30. — The  place  of  anything  is  that  which  is  indicated 


8  INTRODUCTORY  CHAPTER. 

by  the  (true)  response  to  the  question  ''where  ?"  con- 
cerning  that  thing. 

31. — The  aggregate  of  all  conceivable  places  is  called 
Space.  Since  there  is  no  limit  to  our  conception  of 
places,  space  is  limitless  or  infinite. 

32. — Anything  is  said  to  have  extent  if  it  can  be 
conceived  to  be  divided  into  parts  no  two  of  which  occu- 
py the  same  place. 

33. — Distance  is  quantity  of  difference  of  position, 
or  place. 

34. ^Motion  is  appreciable  change  of  place.  Any- 
thing is  said  to  move  when  any  part  of  it  changes  its 
place  through  an  appreciable  distance. 

35. — The  aggregate  of  all  the  different  portions  of 
space  occupied  by  anything  during  its  motion,  is  called 
its  path.  The  path  is  said  to  be  traced  or  generated 
by  the  moving  thing,  which  is  said  to  be  the  generator 
of  its  path. 

36. — Any  body,  t.  e.,  any  limited  portion  of  matter  is 
said  to  be  more  or  less  solid  according  as  (up  to  the 
time  of  yielding  suddenly)  it  offers  more  or  less  resist- 
ance to  pressure  from  opposite  sides  when  otherwise 
unconfined.  Thus,  at  ordinary  temperatures,  a  mass  of 
iron  is  more  solid  than  a  mass  of  lead,  and  that  is  much 
more  solid  than  a  mass  of  paraffme  or  one  of  bees-wax. 

That  property  of  a  body  which  is  more  or  less  perma- 


INTRODUCTORY  CHAPTER.  9 

nent  according  as  the  body  is  more  or  less  solid  is  called 
the  shape  or  form  of  the  body. 

37. — Because  we  consider  any  bounded  part  of  space 
that  we  deal  with  in  geometry  to  keep  the  same  shape 
constantly,  we  call  it  a  geometric  solid  ;  or,  more  fre- 
quently, merely  a  solid,  the  adjective  being  understood 
from  the  nature  of  the  discussion. 

38. — Any  thing  so  small  that  the  distance  between 
no  two  parts  is  appreciable  is  called  a  point. 

39. — The  path  of  a  moving  point  is  called  a  line.  If 
the  line  is  such  that  the  tracing  point  may  return  to  its 
initial  position  without  leaving  the  line  or  retracing  any 
portion  of  it,  the  line  is  a  closed  line;  otherwise  it  is  an 
open  one. 

40. — The  path  of  a  moving  line,  if  other  than  a  line,  is 

called  a  surface.     If  the  surface  completely  encloses 

any  limited  portion  of  space  it  is  a  closed   surface; 

otherwise,  it  is  an  open  one. 

Q.  I.— May  a  line  ever  move  so  as  to  generate  merely  a  line? 
If  so,  show  how. 

41. — The  path  of  a  moving  surface,  if  other  than  a 
surface,  is  a  geometric  body.  * 

42. — Any  two  parts  of  a  line  are  contiguous,  or 
adjacent,  if  they  are  separated  merely  by  a  point ;  any 

*  What  is  here  called  a  geometric  body  is  usually  called  a  solid;  i.  e.,  we  under- 
stand the  word  solid  to  be  restricted  to  mean  a  solid  which  is  neither  line  nor  sur- 
face, unless  the  contrary  is  stated  or  clearly  implied.  Such  will  be  the  usag-e  with 
respect  to  this  word  throughout  the  remainder  of  this  manual.  Lines,  and  sur- 
faces may,  however,  be  very  appropriately  classed  as  solids,  since  their  shapes 
are  supposed  to  be  permanent. 


10  INTRODUCTORY  CHAPTER. 

two  parts  of  a  surface,  if  separated  merely  by  a  line ; 
any  two  parts  of  a  solid  if  separated  merely  by  a  sur- 
face. 

43. — Any  two  points  in  a  line,  a  surface,  or  a  solid, 
are  called  consecutive  *  if  the  distance  between  them 
is  inappreciable.  Saying  that  two  points  of  a  line  are 
consecutive  does  not,  of  course,  mean  that  there  are  no 
other  points  of  the  line  between  them. 

44. — Non-consecutive  points  are  called  separate  * 
points.  When  speaking  of  any  limited  number  of 
points,  separate  points  are  always  to  be  understood 
unless  the  contrary  is  stated  or  clearly  implied. 

45. — The  size  of  a  thing  is  the  quantity  of  its  extent. 
The  size  of  a  line  is  called  its  length  ;  of  a  surface,  its 
area;  of  a  solid,  its  volume. 

Q.  2. — What  is  the  size  of  a  point? 

46. — The  word  size,  as  above  defined,  is  used  only 
with  respect  to  lines,  surfaces,  and  solids.  The  word 
magnitude  is  frequently  used  for  the  word  size,  as 
here  defined,  but  its  use  is  not  restricted  to  the  discus- 
sion of  lines,  surfaces,  and  solids;  e. g.,  we  speak  of  the 
magnitude  of  a  weight,  or  of  a  value,  etc. 

47. — The  four  classes  of  things,  solids,  surfaces,  lines, 
and  points,  are  called  the  geometric  concepts,  because 
they  are  the  things  we  are  continually  thinking  about 
in  every  geometric  discussion.     They   are   all   purely 


*  It  will  be  understood  that  these  words  are  here  used  technically. 


INTRODUCTORY  CHAPTER.  n 

ideal,  or  abstract ;  i.  e.,  as  here  defined,  they  exist  only 
as  mental  conceptions,  and  there  is  nothing  cognizant  to 
the  senses  which  exactly  corresponds  to  them.  Ordi- 
narily, when  we  speak  of  a  solid  we  think  of  some  lim- 
ited portion  of  matter  of  almost  permanent  shape ;  of  a 
surface,  the  outermost  parts  of  a  body  ;  of  a  line,  a  body 
whose  extent  in  one  way  is  much  greater  than  in  oth- 
ers, as,  e.  g.,  a  fish-Iine  ;  of  a  point,  a  very  small  body. 
In  geometry,  however,  the  word  solid  suggests  to  us 
merely  the  idea  of  a  limited  portion  of  space  considered 
with  respect  to  shape  and  to  extent ;  surface,  the  bound- 
ary of  a  solid,  or  that  between  two  contiguous  portions 
of  a  solid,  or  that  which  might  serve  as  such ;  etc.,  etc. 

48. — Geometry  is  that  branch  of  science  which  is 
concerned  in  the  investigation  of  tlie  relations 
of  solids,  surfaces,  and  lines  with  respect  to  their 
two  properties,  extent  and  shape. 

49. — When  any  one  of  these  three  concepts  is  con- 
sidered with  regard  to  its  extent,  it  is  called  a  magni- 
tude ;  with  regard  to  its  shape,  or  form,  a  figure. 
Custom  is  somewhat  variable,  however,  in  the  use  of 
the  word  figure  in  geometry.  When  one  of  the  con- 
cepts is  spoken  of  as  a  figure,  its  shape  is  always 
intended  to  be  considered  and  sometimes  its  extent  also. 
The  context  will  usually  indicate  sufficiently  clearly 
whether  shape  alone  or  both  shape  and  extent  should 
be  considered. 

50.— In  discussing  the  geometric  properties  (extent 


12  INTRODUCTORY  CHAPTER. 

and  form)  of  anything,  the  imagination  frequently 
needs  some  external  aid ;  that  is  to  say,  on  account  of 
the  complexity  of  the  figure  considered,  there  has  to  be 
some  drawing  or  other  thing  to  represent  to  the  mind 
through  the  eye  the  thing  under  discussion.  The  draw- 
ing representing  any  geometric  figure  is  itself  frequent 
ly  called  a  figure.  When  only  lines  and  points  are  rep- 
resented, it  is  usually  called  a  diagram.  Points  are 
usually  represented  by  means  of  dots,  and  named  by 
means  of  letters  (generally  Roman  capitals,)  or  other 
symbols  written  near  them  ;  thus,  this  figure,  '  .a, 
represents  the  three  points.  A,  B,  and  C.  Lines  are 
represented  by  narrow  marks,  and  are  named  by  letters 
(generally  lower-case  Roman  or  Italic)  written  alongside 
them,  or  are  named  by  means  of  their  extremities;  thus, 
this  figure,  - — - — -,  represents  the  line  a,  or  BC.  Sur- 
faces are  represented  by  their  boundary  lines  usually, 
and  the  drawing  representing  them  has  its  light  and  its 
dark  parts  so  distributed,  when  necessary,  as  to  make  a 
sufficiently  complete  picture  to  convey  the  idea  intended. 
A  surface  is  named  by  naming  its  boundary  lines,  or  by 
naming  a  sufficient  number 
of  points  in  it  to  distinguish 
it  from  any  other  surface 
pictured  in  the  same  draw- 
ing. Thus  the  left  hand 
front  surface  (or  face)  of  the  solid  represented  in  this 
drawing,  would  be  named  the  surface  (or  face)  ABC, 


INTRODUCTORY  CHAPTER.  13 

the  right  hand,  the  surface  BCD,  etc.  Solids  are  repre- 
sented and  named  by  means  of  their  surfaces,  or  they 
may  be  named  by  naming  a  sufficient  number  of  their 
points  to  distinguish  them  from  other  solids  represented 
in  the  same  drawing. 

'51. — The  figures  of  geometry  are  treated,  in  the 
drawings  representing  them,  as  being  opaque.  All  lines 
of  the  figure  which  would  be  in  sight  when  the  figure 
has  the  position  represented  in  the  drawing,  are  repre- 
sented (if  at  all)  by  full  lines  (marks);  thus,  . 

Those  which  are  hidden  are  represented  (if  need  be)  by 
means  of  lines  composed  of  dashes  of  medium  length ; 

thus, ;  these  are  drawn  in  the  positions  in 

which  the  lines  they  are  to  represent  would  appear  if 
the  figure  were  transparent.  Lines  not  appearing  in 
the  real  figure  but  drawn  in  the  diagram  to  aid  in  the 
investigation  are  called  auxiliary  lines,  and  are  repre- 
sented by  dashes  alternating  with  dots;  thus, . 

The  lines  sought  for  in  any  discussion  are  called  result- 
ant lines  and  are  represented   by  very  long  dashes; 

thus, . 

52. — The  instruments  which  geometers  have,  through 
common  consent,  restricted  themselves  to  the  use  of  in 
making  their  diagrams,  are  the  pen  or  pencil,  the  ruler, 
and  the  compasses  (an  instrument  used  in  drawing 
circles,  and  in  transferring  lines.)  No  construction  is 
admitted  to  be  legitimately  within  the  range  of  elemen- 
tary geometry,  which  is  composed  of  lines  other  than 


14  INTRODUCTORY  CHAPTER. 

those  which  may  be  drawn  by  aid  of  these  instruments. 
Besides  these  three  instruments,  however,  others  are 
frequently  used  in  practical  drawing,  some  of  which 
serve  merely  to  enable  the  draughtsman  to  do  more 
conveniently  and  expeditiously  what  can  also  be  done 
by  means  of  those  just  mentioned.  These  and  their 
uses  will  be  discussed  later.  Since  they  add  no  new 
powers,  but  act  simply  as  time-savers,  they  may  very 
properly  be  used  in  making  geometrical  constructions. 


BOOK  I. 


RECTILINEAR  FIGURES  IN  A  SINGLE 
PLANE. 


CHAPTER  I. 
LINEAR  RELATIONS. 

53. — The  straight  {i.  e.,  stretched)  line  is  the  line 
whose  position  is  completely  determined  by  the  posi- 
tions of  any  two  of  its  (separate)  points;  that  is  to  say, 
it  is  such  a  line  that  its  position  can  not  be  changed 
without  changing  the  position  of  every  point  in  it  but 
one.  A  fine  fiber  of  silk  suspended  with  a  weight  at 
its  lower  end  suggests  an  approximate  idea  of  a  straight 
line.  Straight  lines  are  also  called  right  lines,  whence 
figures  made  up  of  straight  lines  are  called  rectilinear. 

Q.  3.— Can  you  think  of  any  line  which  is  determined  in  posi- 
tion by  the  position  of  only  one  of  its  points? 

4.  —If  two  or  more  straight  lines  could  be  drawn  joining  the 
same  pair  of  points,  would  it  be  possible  to  say  that  the  straight 
line  is  determined  in  position  by  the  positions  of  two  of  its 
points? 

5.— Immediately  from  the  definition  then,  what  do  we  know 
about  the  number  of  straight  lines  that  may  be  drawn  joining 
the  same  pair  of  points? 


i6  LINEAR  RELATIONS. 

6. — Why  should  the  Hne  now  under  discussion  be  called 
"straight"  (z.  <?.,  stretched)? 

7. — How  does  the  carpenter  get  a  straight  marl<  from  a  point  at 
one  end  of  a  board  to  a  point  at  the  other  end? 

8. — What  is  the  shortest  path  from  one  point  to  another? 

9. — In  what  sort  of  a  line  does  a  ray  of  light  travel? 

10. — What  use  of  the  fact  that  a  straight  line  is  determined  by 
two  points  does  a  hunter  make  in  firing  a  rifle? —  a  farmer  when 
he  wishes  to  set  a  row  of  posts  in  a  straight  line,  or  to  plow  a 
straight  furrow? 

54. — Since  between  two  points  but  one  straight  line 

can  be  drawn,  and  since  the  straight  line  is  the  simplest 

of  all  lines,  we  take  the  length  of  the  straight  line 

joining  two   points  to  be  the   distance   between 

them. 

55. — A  is  said  to  be  nearer  to  B  than  C  is  to  D 
when  the  distance  between  A  and  B  is  smaller  than 
that  between  C  and  D .  Saying  that  C  is  farther  from 
D  than  A  is  from  B  is  equivalent  to  saying  that  A  is 
nearer  to  B  than  C  is  to  D. 

$6. — The    ruler,    rule,    or    straight=edge,    is    an 

instrument  having  an  edge  along  which  straight  lines 
may  be  drawn.  Because  such  an  instrument  is  fre- 
quently called  a  rule,  straight  lines  are  sometimes  called 
ruled  lines. 

57. — When  the  word  line  is  used  hereafter,  through- 
out this  manual,  a  straight  line  is  to  be  understood, 
unless  the  contrary  is  explicitly  stated,  or  unless  the 
context  clearly  indicates  that  the  most  general  meaning 
of  the  word  is  intended.     Moreover,  the  line  is  to  be 


LINEAR  RELATIONS.  17 

understood  to  be  indefinite  in  its  extent  unless  the  con- 
trary is  stated  or  clearly  implied. 

58. — A  line  of  definite  extent  is  called  a  sect.  The 
indefinitely  extended  line  of  which  any  sect  is  a  part  is 
the  seat  of  that  sect.     Sects  of  equal  length  are  equal. 

59. — A  line  of  indefinite  extent  but  having  one  end 
definite  is  a  semi=sect.  Any  given  point  upon  a  line 
evidently  divides  it  into  two  semi-sects. 

60. — Either  of  the  two  semi-sects  into  which  an 
indefinite  line  is  divided  by  any  given  point  upon  it  is 
called  the  complement  of  the  other. 

61. — A  line  not  straight,  but  capable  of  being  divided 
into  parts  of  appreciable  length,  each  of  which  is 
straight,  is  frequently  called  a  broken  line,  and  the 
straight  parts  are  called  its  segments.  We  shall  here 
call  such  a  line  a  chain,  and  its  segments  we  shall  call 
links. 

62. — Three  or  more  points  are  said  to  be  collinear 
when  they  lie  on  the  same  straight  line.  It  is  evident 
that  collinear  points  are  not  independentj  since  the  posi- 
tion of  each  point  is  restricted  (although  not  completely 
determined)  by  the  positions  of  any  other  two  points. 

63. — Two  lines  passing  through  the  same  point  and 
not  coinciding  are  said  to  interse'ct  at  that  point.  Two 
sects  intersect  when  they  have  a  common  point  not  at 
an  extremity  of  either  and  do  not  have  a  common  seat. 

Q.  II.— At  how  many  separate  points  can  two  lines  intersect 


1 8  LINEAR  RELATIONS. 

at  any  one  time? 

12. — If  they  could  intersect  at  two  or  more  points,  could  we 
consistently  say  that  a  straight  line  is  determined  by  two  points.? 

64. — Two  or  more  lines  passing  through  the  same 
point  and  not  coinciding  are  said  to  be  concurrent 
lines.  The  common  point  is  called  the  center  of  con= 
currence.  The  lines  are  called  rays  with  respect  to 
this  center. 

Q.  13.— How  many  rays  may  there  be  through  the  same  center? 

65. — If  a  surface  is  such  that  the  line  through  any 
two  points  of  it  lies  entirely  in  it,  it  is  called  a  plane 
surface,  or  merely  a  plane.  Such  a  surface  is  evident- 
ly perfectly  flat,  and  presents  the  same  appearance 
from  one  side  or  face  as  from  the  other.  Planes  are 
understood  to  be  indefinite  in  extent  unless  the  contrary 
is  stated  or  clearly  implied. 

Q.  14. — How  many  points  are  necessary  to  determine  a  plane? 

15. — Are  ten  collinear  points  sufficient? — five? 

16. — What  is  the  least  number  of  collinear  points  necessary  to 
determine  a  plane? 

17. — Why  should  a  certain  tool  used  by  carpenters  be  called  a 
plane? 

18. — What  kind  of  a  surface  has  a  good  drawing  board? — the 
top  of  a  billiard-table? 

19. — How  does  a  mechanic  work  up  a  set  of  "surface-plates," 
/,  e.,  a  set  of  pieces  of  metal  each  with  a  plane  face,— and 
how  many  of  these  must  he  work  up  in  one  set  in  order  to  work 
up  the  set  independently  of  any  other  set? 

20.— If  two  planes  intersect  {i.  e.,  cut  each  other)  what  sort  ot  a 
line  is  the  line  of  intersection? 

21. — How  does  the  carpenter  test  the  surface  of  a  piece  which 
he  has  planed  to  see  whether  or  not  it  is  perfectly  flat? 


LINEAR  RELATIONS.  19 

22.— How  does  a  stone-cutter  find  out  what  parts  of  a  stone  to 
dress  off  if  he  wants  to  give  the  stone  a  flat  face? 

66. — Tlie  trace  of  one   figure   upon   another   is 

the  part  common  to  the  two.  It  is  understood  when 
we  speak  of  the  trace  of  one  figure  upon  another  that 
each  has  parts  not  included  by  the  other. 

Q.  23.— What  is  the  trace  of  one  line  upon  another? 

24.— Of  a  line  upon  a  plane? 

25. — Of  one  plane  upon  another? 

67. — Any  figure  all  of  whose  parts  lie  in  a  single 
plane  is  called  a  plane  figure.  On  account  of  the 
greater  simplicity  of  plane  figures  and  of  the  fact  that  a 
knowledge  of  many  of  their  properties  is  necessary  to  a 
discussion  of  those  of  non-plane  figures,  the  study  of 
plane  figures  is  undertaken  first. 

68. — To  make  a  geometric  drawing  (/.  ^.,  one  in 
which  the  extent  and  shape  of  the  various  figures  must 
be  exact,)  the  instruments  before  mentioned*  are  usual- 
ly necessary,  and  a  good  drawing  board  also  if  the 
drawing  is  to  be  made  upon  a  thin  sheet  of  paper. 

The  paper  is  mounted  (or  fastened)  upon  the  board 
usually  by  tacking  down  its  edges  or  else  by  pasting 
them  to  the  board.  In  case  the  edges  are  to  be  tacked 
down  it  is  advisable  to  use  draughtsman's  thumb-tacks, 
which  are  broad,  flat-headed  tacks,  easily  pressed  into 
soft  wood  by  the  thumb,  and  whose  heads  being  flat 
and  thin  do  not  materially  interfere  with  the  use  of  the 

■■'•  See  art.  52,  page  13. 


20  LINEAR  RELATIONS. 

draughting  instruments.  In  case  the  edges  are  to  be 
pasted  to  the  board,  a  margin  from  one-half  to  three- 
quarters  of  an  inch  wide  should  be  folded  up  all  around 
the  sheet,  and  the  rest  of  the  paper  thoroughly  damp- 
ened. The  water  for  this  purpose  must  be  entirely 
clean,  else  the  paper  will  be  streaked.  The  dampening 
being  completed,  the  dry  margins  are  to  be  pasted  to 
the  board,  being  careful  in  pasting  to  stretch  the  damp- 
ened paper  sufficiently  to  make  it  dry  perfectly  flat. 
For  pasting  the  dry  margins  to  the  board,  strong  and 
quickly  drying  paste  or  glue  must  be  used,  as  the  edges 
must  be  securely  fastened  before  the  paper  has  dried 
materially,  else  it  will  wrinkle.  Great  care  must  be 
taken  not  to  wet  these  margins,  as  wetting  them  will 
cause  them  to  tear.  In  order  to  stretch  the  paper  to  the 
best  advantage  the  second  margin  parted  down  should 
be  the  one  opposite  to  the  one  first  pasted.  This  second 
method  of  mounting  the  paper  upon  the  board  should 
be  used  whenever  it  is  to  remain  for  any  considerable 
time  on  the  board,  as  in  making  an  extensive  and  com- 
plicated drawing;  also  whenever  wet-tints  or  ink  wash- 
es are  to  be  used.  These  would  cause  the  paper  to 
wrinkle  in  drying  if  it  were  mounted  without  having 
been  previously  moistened  and  stretched.  For  draw- 
ings such  as  will  be  required  by  most  of  the  exercises  in 
this* manual,  it  will  be  sufficient  to  secure  the  paper  by 
thumb-tacks. 

For  drawing  straight  lines  the  pencil  should  be  given 


LINEAR  RELATIONS.  21 

a  point  like  a  needle  point,  or  else  the  marking  portion 
should  be  made  wedge-shaped:  this  latter  form  we  shall, 
for  convenience,  call  a  **wedge-point"  although  it  is  not, 
properly  speaking,  a  point  at  all.  For  keeping  the  pen- 
cil sharp  a  piece  of  very  fine  sand-paper  or  a  very  fine 
file  is  convenient.  The  pencil  used  should  always 
be  hard  enough  to  make  a  fine,  clean  mark,  but  not  so 
hard  as  to  indent  the  paper  very  considerably,  else  the 
marks  will  be  difficult  to  erase  should  any  erasure  be 
necessary.  For  drawing  straight  lines,  the  wedge-point 
is  usually  more  satisfactory  than  the  needle-point, 
because  it  demands  less  frequent  sharpening. 

In  drawing  lines  along  the  ruler,  care  must  be  taken 
that  the  pencil  (or  ruling-pen,  as  the  case  may  be)  is 
not  tilted  one  way  or  the  other.  Likewise  in  drawing 
lines  with  the  compasses,  care  must  be  taken  that  the 
portions  of  the  legs  next  the  paper  are  not  tilted  one 
way  or  the  other  to  it.  A  good  pair  of  compasses  is 
always  provided  with  a  joint  in  each  leg. 

For  erasing  pencil  marks,  soft,  clean,  uniform  india- 
rubber  should  be  used.  For  erasing  ink  marks,  a 
harsher  grade  of  rubber  is  sometimes  used;  more  fre- 
quently, however,  a  steel  eraser  is  first  used,  then  india- 
rubber,  and  the  erased  surface  finally  burnished.  Very 
fine  sand -paper  often  gives  very  satisfactory  results. 
A  dense,  tough  paper  must  be  used  where  much  eras- 
ing is  likely  to  be  necessary. 

69. — To  test  the  straightness  of  the  edge  of  the 


22  LINEAR  RELATIONS. 

ruler,  take  two  points,  A  and  B,  on  the  mounted  sheet, 
nearly  as  far  apart  as  the  ruler  is  long,  or,  if  this  be 
impossible,  as  far  apart  as  the  size  of  the  sheet  of  paper 
will  conveniently  permit.  Along  the  edge  of  the  ruler 
which  it  is  desired  to  test,  draw  a  line  through  these  two 
points.  Then,  calling  the  point  of  the  ruler  nearest  A, 
A',  and  that  nearest  B,  B',  turn  the  ruler  over  upon  its 
opposite  face,  keeping  A'  at  A  and  B'  at  B,  and  draw 
another  line  through  A  and  B  along  the  edge  to  be 
tested.  If  the  two  lines  so  drawn  coincide  throughout, 
the  tested  edge  of  the  ruler  is  straight  ( ? ) ;  if  they  do 
not,  it  is  not  straight  {?). 

Q.  26.— Why  should  the  points  A  and  B  be  taken  as  far  apart 
as  is  practicable  ? 

27. — In  actual  drawing  can  a  straight  line  (mark)  be  deter- 
mined so  exactly  by  two  points  (dots)  a  fiftieth  of  an  inch  apart, 
as  by  two  five  inches  apart.? 

28.— If  so,  why,  if  not,  why  not? 

29.— If  the  ruler  in  the  second  position  cover  the  line  first 
drawn  between  A  and  B,  what  portion  of  the  ruler  should  be  cut 
away  in  order  to  straighten  it,  and  how  much  of  it  with  respect 
to  the  distance  between  the  two  lines? 

70. — One  sect  a  is  added  to  another  sect  d  by  pro- 
ducing (/.  ^.,  drawing  out,  extending)  d  until  the  pro- 
duced part  equals  a.  It  is,  of  course,  understood  that 
the  produced  part  must  make  with  d  a  single  sect  and 
not  a  two-linked  chain. 

71. — The  sect  so  obtained,  the  first  part  of  which  is  d, 
the  second  equal  to  a,  is  the  sum  of  the  two. 

72.— Three  or  more  things  are  added  by  adding  to  the 


LINEAR  RELATIONS.  23 

sum  of  the  first  two  the  third,  to  the  sum  of  the  first 

three  the  fourth,  etc. 

73. — The  sign  of  addition  is  the  same  as  in  arithmetic 

and  in  algebra,  and  signifies  here  as  there  that  the  thing 

whose  symbol  follows  the  sign  is  to  be  added  to  that 

whose  symbol  precedes  it. 

Q.  30. — What  relation  exists  between  the  length  of  the  sum  of 
two  sects  and  the  sum  of  their  lengths? 

74. — Sects  are  transferred  from  one  place  in  a  draw- 
ing to  another  by  means  of  the  compasses  or  dividers. 
The  legs  of  the  dividers  are  spread  until  when  one  point 
rests  on  one  end  of  the  sect  the  other  point  may  rest  on 
the  other  end.  The  distance  between  the  points  is 
then  equal  to  the  length  of  the  sect,  and  by  careful 
handling  of  the  dividers  this  length  may  be  set  off  upon 
any  desired  line.  In  case  great  accuracy  is  required,  or 
in  case  doubt  exists  concerning  the  constancy  of  the  dis- 
tance between  the  points,  the  dividers  should  again  be 
applied  to  the  first  sect,  noticing  whether  or  not  the  dis- 
tance between  the  points  is  yet  equal  to  the  length  of 
the  sect.  Opening  the  legs  of  the  dividers  so  that  the 
distance  between  the  points  equals  the  length  of  the 
sect  PQ  is  called  "taking  the  sect  PQ  in  the  dividers." 

E.  31.— Take  four  sects  a,  b,  c,  and/,  no  two  being  equal  and 
find  their  sums,  taking  them  in  various  orders,  twenty-four  in  all. 
What  relation,  if  any,  exists  among  the  sums  so  found? 

75. — The  problem,  to  find  the  ratio  between 
two  commensurable  sects,  evidently  consists  of  two 
parts;  first,  to  find  a  common  measure  or  common  unit  of 


24  LINEAR  RELATIONS. 

the  two  sects,  -  and  second,  to  find  the  lengths  of  these 
sects  in  terms  of  the  common  measure  and  take  their 
ratio. 

First,  to  find  the  common  measure.     If  the  sects  are 
unequal,  suppose  the  longer  called  AB,  and  the  shorter 

^— — ^/ 

J,. . *— r» 


CF.  Taking  CF  in  the  dividers,  lay  it  off  as  many 
times  as  possible  on  AB,  beginning,  say,  at  the  end  A. 
If  the  line  AB  is  a  multiple  of  CF,  CF  is  the  measure 
sought ;  if  not,  the  last  point  of  division  will  fall  some- 
where between  B  and  A  and  nearer  to  B  than  C  is  to  F. 
Call  this  point  G.  Apply  GB  in  like  manner  to  CF, 
beginning,  say,  at  C,  and  if  CF  is  not  a  multiple  of  GB, 
call  the  last  point  of  division  on  CF,  H ;  and  in  like 
manner  apply  HF  to  GB,  and  so  on  back  and  forth  until 
some  remainder  is  found  which  is  a  sub-multiple  of  the 
last  preceding  remainder ;  this  is  the  common  measure 
sought,  for  each  remainder  and  each  divisor  before  it  is 
a  multiple  of  it,  and  so  their  sums  must  be. 
A  Second,  to  find  the  ratio  between  the  sects,  express 
the  length  of  each  divisor  in  terms  of  the  common  meas- 
ure, after  which  the  lengths  of  CF  and  AB  may  easily 
be  expressed  in  terms  of  it.  The  ratio  between  these 
lengths  is  the  ratio  required. 

Q.  and  E.  32.— What  processes  of  arithmetic  and  algebra  is 
this  operation  essentially  similar  to  ? 


LINEAR  RELATIONS.  25 

33.— Draw  three  sects  at  random  and  find  their  common 
measure. 

^6. — It  is  apparent  that,  since  the  successive  remain- 
ders in  the  operation  just  described  become  smaller,  we 
shall  in  any  given  case  in  practice  soon  find  a  remainder 
so  small  that  we  are  unable  to  deal  with  it  on  our  draw- 
ing. All  pairs  of  sects  will  thus  in  practice  appear  to  be 
commensurable.  We  become  aware  of  incommensura- 
ble magnitudes  only  through  calculation.  Later  this 
will  appear  more  clearly,  especially  in  studying  trigo- 
nometry. 

77, — In  practice,  we  are  most  frequently  concerned 
with  the  ratio  a  sect  bears  to  a  customary  standard  of 
length,  as  the  inch  in  the  English  system,  the  centime- 
ter in  the  French,  etc.  This  is  so  frequently  called  for 
that  we  find  it  convenient  to  have  a  good  collection  of 
the  multiples  and  sub-multiples  of  the  unit  marked  off 
on  an  instrument  which  is  called  a  scale.  All  that  we 
need  do  then  to  find  the  length  of  a  sect  in  terms  of  the 
customary  unit  is  to  take  the  sect  in  the  dividers  and 
apply  it  to  the  graduated  (i.  e.,  measured  and  marked) 
line  of  a  suitable  scale  and  observe  how  many  units  and 
what  fraction  of  a  unit  it  covers.  Practically,  too,  when 
we  wish  to  find  the  ratio  between  two  sects  we  find 
their  lengths  by  means  of  some  suitably  graduated 
scale,  and  then  compute  the  ratio  of  one  of  these  lengths 
to  the  other.  This  is  much  more  convenient  than  the 
method  first  described,  and  in  most  cases  at  least  as 
nearly  accurate. 


26  LINEAR  RELATIONS. 

78.-T0 1  tHsect }  «  ^^^^  's  to  divide  it  into  |  Jj^"^^ 
equal  parts.  That  point  of  a  sect  which  bisects  it  is  its 
middle  point.  In  practice,  especially  where  a  sect  is 
to  be  divided  into  only  a  small  number  of  equal  parts 
and  where  only  a  moderate  degree  of  accuracy  is 
desired,  we  frequently  divide  it  into  equal  parts  by  trial. 
The  dividers  *  are  opened  until  it  is  estimated  that  they 
contain  between  their  points  the  proper  distance,  and 
the  sect  between  their  points  is  laid  off  upon  the  given 
sect  the  desired  number  of  times.  Should  the  sect  be 
more  than  covered  the  points  of  the  dividers  are  too  far 
apart ;  and  contrarily.  A  little  experience  will  suffice  to 
show  that  this  is  a  very  tedious  method  when  the  sect 
is  to  be  divided  into  a  considerable  number  of  parts. 
As  was  said  at  the  beginning,  this  method  is  a  method 
"by  trial."     An  exact  method  will  appear  later. 

79. — A  line  is  said  to  be  curved  when  no  part  of  it 
having  appreciable  length  is  straight.  A  curved  line  is 
usually  called  by  the  briefer  name,  a  curve. 

80. — Any  open  curve  is  called  an  arc. 

81. — A  sect  joining  any  two  points  of  a  curve  is  called 
a  chord.  When  tke  chord  of  any  arc  is  spoken  of,  the 
chord  joining  the  extremities  of  the  arc  is  meant. 

*  The  use  of  the  dividers  in  dividing  sects  into  equal  parts  and  in  dividing:  one 
sect  by  another  as  in  the  process  of  finding  the  common  measure  between  two 
sects  shows  the  reason  for  the  name.  Hitherto  the  words  dividers  and  compass- 
es have  been  used  interchangeably.  Hereafter  an  instrument  for  transferring 
lengths  and  for  dividing  sects  will  be  called  a  pair  of  dividers.  A  similar  instru- 
ment used  for  drawing  will  be  called  a  pair  of  compasses.  Both  points  of  the 
dividers  should  be  needle  points.  One  of  the  compass-points  should  be  pen,  pen- 
cil, or  crayon. 


LINEAR  RELATIONS.  27 

82. — A  plane  curve  all  of  whose  points  are  equally 
distant  from  a  fixed  point  in  the  plane  is  called  a  circu- 
lar arc,  or,  if  a  closed  curve,  it  is  called  a  circle.  The 
circular  arc  being  the  simplest  of  all  arcs  and  the  most 
frequently  used,  the  word  arc  is  always  to  be  under- 
stood as  meaning  circular  arc  unless  some  other  meaning 
is  stated  or  clearly  implied. 

Q.  34. — With  what  instrument  are  circular  arcs  usually  drawn? 

83. — The  fixed  point  in  the  plane  of  a  circular  arc  or 
circle  from  which  all  points  of  the  curve  are  equidistant 
is  called  the  center  of  the  arc  or  circle. 

Q.  35.— How  many  centers  can  a  circle  have? 

84. — That  point  of  an  arc  which  bisects  it  {i.  e., 
divides  it  into  two  equal  parts)  is  called  its  middle 
point. 

85. — A  sect  drawn  from  the  center  of  an  arc  or  circle 

to  any  point  of  it  is  called  a  radius  (pi.  radii.)     When 

we  speak  of  the  radius  of  any  arc  we  mean  the  length  of 

a  radius  of  that  arc. 

Q.  36. — What  relations,  if  any,  exist  between  the  radii  of  any 
arc? 

86. — To  strike  an  arc  with  a  certain  sect  as  radius 
is  to  draw  one  whose  radius  is  the  length  of  that  sect. 

%7. — A  chord  of  a  circle  passing  through  the  center  is 
a  diameter  of  that  circle. 

Q.  37.— What  relation,  if  any,  exists  between  the  diameters  of 
a  circle  ?— between  a  diameter  and  a  radius  of  the  same  circle  ?^ 
F.  38.     Draw  a  circle  and  show  a  diameter,  a  radius,  a  chord. 


28  LINEAR  RELATIONS. 

39.— Draw  an  arc  with  its  chord. 

88. — The  locus  (pi.  loci)  of  a  point  satisfying  a  given 
condition  is  the  figure  all  of  the  points  of  which  and  no 
points  outside  of  which  satisfy  the  given  condition.  If 
we  are  required  to  find  a  point  satisfying  two  condi- 
tions, we  find  the  points  common  to  the  loci  of  points 
satisfying  those  two  conditions.  Such  points  being  on 
both  loci  evidently  satisfy  both  conditions. 

89. — To  '»construct''  a  locus  we  locate  a  sufficient 
number  of  points  upon  it  to  give  us  the  idea  of  its 
appearance,  and  join  these  points  by  a  suitable  line  or 
other  figure.  Of  course  the  more  points  that  are  located 
exactly,  the  more  accurate  will  our  construction  be. 

Q.  and  E.  40. — Construct  the  locus  of  a  point  which  shall  be 
at  a  distance  of  two  inches  from  a  given  point. 

41. — What  kind  of  figure  is  the  locus  above  called  for,  and 
what  relation  does  the  given  point  bear  to  it? 

42. — Find  a  point,  P,  which  shall  be  at  a  distance  of  two  inch- 
es from  one  of  two  given  points,  A,  and  three  inches  from  the 
other,  B. 

43.— How  many  solutions  [i.  <?.,  how  many  points  satisfying 
the  requirements)  when  A  and  B  are  six  inches  apart?— five  and 
a  quarter  ?— five? — two  and  a  half?— one  inch?— one  half  of  an 
inch?    Make  constructions  showing  these  various  cases. 

44.— Construct  the  locus  of  a  point  which  shall  be  equi-distant 
from  each  of  two  given  points. 

45. — What  kind  of  figure  is  it? 

46. — Does  the  shape  or  size  of  this  locus  vary  with  the  posi- 
tions of  the  two  points  ? 

47.— Does  its  position  ? 

48.— Construct  the  locus  of  a  point  which  shall  be  twice  as  far 
from  one  of  two  given  points  as  from  another. 


LINEAR  RELATIONS.  29 

49. — What  kind  ot  figure  is  it? 

50.— What  relation,  if  any,  exists  between  its  size  and  the  dis- 
tance between  the  given  points? 

51.— Same  for  a  point  which  shall  be  three  times  as  far  trom 
one  of  two  given  points  as  from  the  other; — four  times; — five 
times. 

52. — What  general  relation  exists  among  the  loci  obtained  in 
Ex.  48  and  51  ? 

90. — Two  semisects  terminating  at  the  same  point 
form  a  figure  called  an  angle.  The  two  semisects  are 
its  sides  or  arms,  and  are  said  to  **incIose**  the  angle. 
The  point  common  to  the  two  sides  is  called  the  vertex 
of  the  angle. 

91. — The  essential  part  of  the  idea  of  an  angle 
is  that  of  its  shape  at  the  vertex.  The  lengths  of 
the  sides  being  indefinite  are  not  taken  into  considera- 
tion. 

92. — If  the  two  sides  of  an  angle  are  complements 
(see  art.  60,  page  17, )  of  each  other,  the  angle  is  called 
a  straight  angle. 

93. — It  .will  be  noticed  that  the  two  sides  of  any 
angle  not  straight  (nor  yet  a  zero  angle, — see  art.  108, 
page  32, )  are  sufficient  to  determine  a  plane  in  position. 
The  plane  so  determined  is  called  the  plane,  or  the 
seat,  of  the  angle. 

94. — The  plane  of  an  angle  is  divided  into  two  parts 
by  the  sides  of  the  angle.  Each  of  these  is  frequently 
called  an  angle,  and  this  is  the  meaning  with  which  the 
word   will   be   used  throughout  the  remainder  of   this 


30  LINEAR  RELATIONS. 

manual  unless  some  other  is  indicated. 

95. — Of  the  two  angles  having  the  same  sides,  the 
one  which  is  crossed  by  the  sect  joining  any  point  in 
one  side  to  any  point  in  the  other  is  called  a  convex 
angle ;  the  other  is  called  a  concave  angle. 

96.— An  angle  is  divided  into  parts  by  a  semi-sect 
(or  semi-sects)  lying  in  it  and  drawn  from  its  vertex. 
It  will  be  noticed  that  the  parts  of  angles  are  also  angles. 

Q.  53.— Of  what  kind  are  the  parts  of  a  convex  angle?— of  a 
straight  angle? — of  a  concave  angle? 

54.— Which  is  the  larger,  a  convex  angle  or  a  straight  angle? 
— a  straight  angle  or  a  concave  angle? 

97. — An  angle  is  named  by  naming  its  vertex  or  by 
naming  its  sides.  In  case  two  or  more  angles  have  a 
common  vertex,  it  evidently  is  insufficient  to  name  the 
vertex ;  in  this  case  the  two  sides  are  named,  or  else  the 
vertex  is  named  between  two  other  points  one  of  which 
pertains  to  one  side,  the  other  to  the  other  side.  It 
should  be  particularly  noticed  that  when  an  angle  is 
named  by  means  of  three  points,  the  vertex  is  custom- 
arily named  between  the  other  two.  When  any  angle 
is  named,  of  the  two  angles  to  which  the  name  applies, 
the  convex  is  always  to  be  understood  unless  the  con- 
trary be  indicated  or  unless  the  angle  be  straight. 

98. — Two  angles  are  adjacent  when  they  have  such 
positions  that  they  are  the  two  parts  into  which  a  third 
angle  is  divided  by  their  common  side. 

99. — Two  angles  are  vertical  to  each  other  when 


I 


LINEAR  RELATIONS.  31 

the  sides  of  one  are  the  complements  (see  art.  60,  page 
17,)  of  those  of  the  other. 

100. — Two  angles  are  equal  when  upon  being 
applied  one  to  the  other  so  that  they  have  a  side  and 
the  vertex  of  one  coinciding  with  a  side  and  the  vertex 
of  the  other,  the  other  two  sides  will  coincide.  It  is  to 
be  understood  that  if  one  is  convex  the  other  is  convex 
also. 

loi. — One  angle  is  added  to  another  by  being 
placed  adjacent  to  it.  The  angle  of  which  the  two  giv- 
en angles  then  form  parts  is  the  sum  of  the  two. 

102. — An  angle  is  bisected  when  it  is  divided  into 
two  equal  parts;  in  fact,  to  bisect  any  figure  is  to  divide 
it  into  two  equal  parts ,  to  trisect  it  is  to  divide  it  into 
three  equal  parts,  etc.  The  line  bisecting  any  angle  is 
called  the  bisector  of  the  angle. 

103. — Half  a  straight  angle  is  called  a  right  angle, 
or  an  orthogon.  The  right  angle  is  one  of  the  most 
commonly  used  units  of  angle  in  geometry. 

104. — The  most  commonly  used  unit  of  angle,  how- 
ever, is  the  degree,  which  is  the  ninetieth  part  of  a 
right  angle.  The  degree  is  subdivided  into  sixty  equal 
parts,  each  of  which  is  called  a  minute;  and  each  min- 
ute is  further  subdivided  into  sixty  equal  parts,  each  of 
which  is  called  a  second.  Seconds  are  sometimes  each 
subdivided  into  sixty  equal  parts,  each  of  which  is  called 
a  third  ;  but  the  subdivisions  of  the  second  are  usuallv 


32  LINEAR  RELATIONS. 

decimal.  The  symbols  for  degrees,  minutes  and  sec- 
onds are  %  ',  " ;  thus  2f  15'  42". 3  is  read  27  degrees, 
15  minutes,  and  42.3  seconds. 

105. — Any  angle  smaller  than  a  right  angle  is  called 
an  acute  angle;  one  larger  than  a  right  angle  and 
smaller  than  a  straight  angle  is  called  an  obtuse  angle. 
Acute  and  obtuse  angles  are  called  oblique  angles. 

f  right  angle    ^ 
106. — If  the  sum  of  two  angles  is  a^  straight  angle  |^ 

[  full  plane        J 

fcomplementl 

either  is  said  to  be  theJ  supplement  ^of  the  other. 
[explement    J 

107. — An  angle  and  its  adjacent  explement  are  said 
to  be  conjugate. 

108. — Two  coincident  semi-sects  are  sometimes  said 
to  include  a  zero  angle.  With  this  usage  agreed  to,  it 
may  be  said  that  any  two  semi-sects  drawn  from  the 
same  point  include  an  angle. 

109. — If  one  of  the  four  convex  angles  formed  at  the 
point  of  intersection  of  two  lines  is  a  right  angle,  the 
lines  are  said  to  be  perpendicular,  normal,  or 
orthogonal,  to  each  other. 

no. — Two  sects  or  two  semisects  or  a  sect  and  a 
semisect  are  said  to  be  perpendicular  to  each  other  when 
their  seats  are  so. 

To  erect  a  perpendicular  to  a  given  line 


'  To  drop  a  perpendicular  to  a  given  line 


LINEAR  RELATIONS.  33 

at  a  given    point    upon   that   line"! 

X  •  •  X      -^u     ^  ^u  ..  1-      fis  to  draw  through 

from  a  given  point  without  that  hne  J 

the  given  point  a  perpendicular  to  the  given  line.     The 

foot  of  a  perpendicular  is  the  point  where  it  meets 

the  base,  i.  e.,  the  line  to  which  it  is  perpendicular ;  or 

it  is  the  point  at  which  it  would  meet  the  base  if  both 

were  sufficiently  produced. 

112. — Any  sect  extending  from  a  point  outside  a  line 
to  a  point  of  that  line  is  called  an  oblique  if  it  malos 
with  that  line  an  oblique  angle. 

113. — The  orthogonal  projection  of  a  point  upon  a 
line  is  the  foot  of  the  orthogonal,  or  perpendicular,  from 
the  point  to  the  line.  The  orthogonal  projection  of  a 
figure  is  the  aggregate  of  the  orthogonal  projections  of 
its  points.  Other  sorts  of  projection  are  used  as  well  as 
orthogonal,  but  projection  is  understood  to  be  orthogonal 
unless  some  other  is  stated  or  implied. 

114. — A  figure  composed  of  three  (independent) 
points  and  their  connecting  sects  is  called  a  triangle, 
or  a  trigon,  or  a  trilateral. 

1 1 5.— The  three  points  are  called  the  vertices  of  the 
trigon;  the  connecting  sects  its  sides;  the  angles  with- 
in the  figure  and  having  its  vertices  for  their  vertices, 
its  angles. 

116. — The  side  upon  which  the  trigon  is  supposed  to 
rest  is  called  the  base.  The  other  two  sides  are  the 
legs.     When  we  speak  of  the  vertex  of  a  trigon,  the 


34  LINEAR  KELATIONS. 

one  opposite  the  base  is  understood.     By  the  vertical 
angle  of  a  trigon  is  meant  the  one  at  the  vertex. 

117. — Customarily  when  the  word  trigon,  triangle,  or 
trilateral  is  used  we  understand  it  to  mean  besides  the 
vertices  and  sides  the  portion  of  a  plane  enclosed  by 
the  three  sides. 

118. — With  respect  to  their  sides,  trigons  are  classi- 
fied as  scalene  (unequal  legged)  having  no  two  sides 
equal;  isosceles  (equal  legged)  having  two  sides  equal ; 
and  equilateral,  having  all  three  sides  equal :  with 
respect  to  their  angles, — as  acute,  each  angle  acute; 
right,  one  angle  right ;  obtuse,  one  angle  obtuse ;  and 
equiangular,  isogonic,  or  isogonal,  having  the  three 
angles  equal.  In  every  isosceles  trigon,  the  two  equal 
sides  are  to  be  taken  as  the  legs  unless  the  contrary 
is  stated  or  implied ;  in  a  right  trigon,  the  two  sides 
including  the  right  angle.  The  side  opposite  the  right 
angle  in  any  right  trigon  is  called  the  hypotenuse. 

119. — The  sides  of  a  trigon  are  usually  represented 
by  the  three  letters  a,  b,  and  c\  the  vertices  opposite  by 
the  letters  A,  B,  and  C  respectively;  and  the  angles  at 
these  vertices  by  a,  /?,  and  y  *,  When  two  or  more  tri- 
gons are  to  be  denoted,  subscripts  or  indices  are  used  ; 
thus  the  sides  of  the  first  may  be  represented  by  a^,  b^, 
^1,  and  those  of  the  second  by  ^2»  <^2»  <^2>  ^tc. 


*  «,  />.  and  /  are  the  lower-case  forms  of  the  first  three  letters  of  the  Greek 
alphabet.     Their  names  are  respectively  alpha,  beta,  and  gamma. 


LINEAR  RELATIONS.  35 

120 — The  face  of  any  plane  figure  which  is  toward 
the  eye  of  the  draughtsman  as  the  figure  is  being 
drawn,  we  shall  call  the  obverse;  the  opposite  face, 
the  reverse. 

121. — In  plane  geometry,  in  saying  that  two  figures 
are  equal  we  mean  that  the  obverse  of  one  is  equal  to 
the  obverse  of  the  other.  If  the  obverse  of  one  is  equal 
to  the  reverse  of  the  other,  we  shall  say  the  figures  are 
opposite.  For  the  sign  of  opposition  we  shall  use  ^  ; 
^  3  ^  will  be  read,  a  is  the  opposite  of  b.  Unequal  will 
be  used  to  mean  neither  equal  nor  opposite. 

in  the  exercises  of  this  and  the  next  article  the  fig- 
ures may  be  cut  out  of  the  drawing  for  purposes  of 
comparison,  or  they  may  be  drawn  on  thin  transparent 
paper,  such  as  tracing  paper,  or  the  so-called  "onion- 
skin" note-paper.  The  latter  mode  will  be  the  more 
expeditious  and  satisfactory. 

Q.  and  E.  55. — If  one  angle  is  the  opposite  of  another,  are  the 
angles  equal  or  unequal  1 

56. — Can  an  isosceles  trlgon  be  the  opposite  of  a  scalene 
trigon? 

57. — If  one  scalene  trigon  is  the  opposite  of  another  are  the 
two  equal? 

58. — Same  for  isosceles. 

59. — Same  for  equilateral. 

60.— If  two  angles  of  a  trigon  are  equal,  is  the  trigon  equal  to 
its  opposite? 

61.— Is  it  if  no  two  angles  are  equal  ? 

62.— Is  it  if  it  is  isogonic?  ' 

122. — To  construct  a  trigon  whose  sides    shall 


36  LINEAR  RELATIONS. 

be  equal  to  given  sects.  Suppose  the  given  sects 
m,  71,  and  /.  Draw  a  sect  AB  equal  to  one  of  the  given 
sects,  say  m,  and  then  construct  the  locus  of  a  point 
whose  distance  from  A  is  the  length  of  another,  say  n, 
and  the  locus  of  a  point  whose  distance  from  B  is  the 
length  of  the  third  sect,/.  Any  point  common  to  these 
loci  may  evidently  serve  as  the  third  vertex,  C,  of  the 
required  trigon,  which  is  finished  by  drawing  the  sects 
AC  and  BC. 

Q.  and  E.  63. — What  relation  must  exist  among  the  lengths  of 
the  three  sides  of  a  trigon  ? 

64. — Can  a  trigon  be  constructed  the  lengths  of  whose  sides 
shall  be  3,  5,  and  10  inches?  If  so,  construct  one;  if  not,  say 
why  not? 

65.— Can  one  be  constructed  with  sides  3  inches,  5  inches,  and 
8  inches  long  respectively? — 3  inches,  5  inches  and  anything  less 
than  8  inches?— 3  inches,  5  inches,  and  2  inches?— 3  inches,  5 
inches,  and  iK  inches? 

66.— Construct  a  scalene  trigon, — an  isosceles  trigon, — an 
equilateral  trigon. 

67. — What  relation  exists  among  the  angles  of  the  scalene  tri- 
gon? Are  any  two  of  them  equal?  Which  two,  if  any,  are 
equal?  If  no  two  are  equal,  which  side  is  the  smallest  angle 
opposite? — which  the  largest?  Are  the  angles  proportional  to 
the  sides  opposite? 

68. — Same  for  isosceles  trigon. 

69. — Same  for  equilateral  trigon. 

7o.--Try  several  different  and  unequal  scalene  trigons,  and  see 
if  your  answers  to  question  number  67  hold  true  of  these  also. 

71. — What  general  relation,  if  any,  seems  to  hold  between  the 
relations  among  the  sides  of  a  trigon  and  the  relations  among 
the  angles  opposite  them? 

72. — If  the  sides  of  one  scalene  trigon  are  respectively  equal  to 
those  of  another,  what  relation  if  any  exists  between  the  two? 


LINEAR  RELATIONS.  37 

73. — Same  for  isosceles  trigons. 

74.— Same  for  equilateral  trigons. 

75.— Construct  two  equilateral  trigons  in  which  the  sides  of 
one  are  smaller  than  those  of  the  other.  What  relation,  if  any, 
exists  between  the  angles  of  one  and  those  of  the  other?  If  the 
angles  are  unequal  in  which  are  they  the  larger? 

76.— After  making  the  comparisons  called  for  in  this  set  of 
exercises,  find  the  sum  of  the  angles  of  each  trigon  you  have 
used.  Is  there  any  uniformity  in  these  various  sums?  If  so, 
what? 

77. — Can  a  trigon  have  two  right  angles? — two  obtuse  angles? 
—only  one  acute  angle? 

123. — The  sect  joining  any  vertex  of  a  trigon  to  the 
middle  point  of  the  side  opposite  is  called  the  median 
of  the  trigon  to  that  side. 

124. — The  sect  bisecting  any  angle  of  a  trigon  and 
terminating  in  the  side  opposite  is  called  the  bisector  of 
the  trigon  to  that  side. 

125. — The  sect  drawn  from  any  vertex  of  a  trigon 
perpendicular  to  the  side  opposite  and  terminating  in 
that  side  (or  that  side  produced)  is  called  the  altitude 
of  the  trigon  upon  that  side. 

Q.  and  E.  78.— Draw  three  or  more  unequal  scalene  trigons, 
and  draw  the  three  medians  of  each.  What  relation,  if  any, 
exists  among  the  three  medians  of  any  one  of  these  trigons? 

79. — Do  the  same  with  isosceles  trigons  and  with  equilateral 
trigons.  Do  your  conclusions  in  exercise  number  78  hold  true 
with  these  trigons  also?  If  not,  what  modifications  are  neces- 
sary ? 

80. — What  peculiar  relation,  if  any,  does  the  median  to  the 
base  of  an  isosceles  trigon  bear  to  the  base  ? 

81.— What  relation  does  it  bear  to  the  vertical  angle? 

82.— Devise  some  means  of  erecting  a  perpendicular  at  a  given 


38  LINEAR  RELATIONS. 

point  of  a  given  line,  making  use  of  your  conclusion  in  q.  no.  80. 
83. — How  many  different  perpendiculars  can  be  erected  at  the 
same  point  and  on  the  same  side  of  the  line?    How  might  this 
have  been  known  from  the  definition  of  a  perpendicular? 

84. — Devise  some  means  of  dropping  a  perpendicular  to  a  line 
from  a  point  outside  the  line,  making  use  of  your  conclusion  in 
q.  no.  80. 

85.— How  many  different  perpendiculars  can  be  dropped  upon 
the  same  line  from  the  same  point  outside  the  line? 

86.— Devise  some  means  of  finding  exactly  the  middle  point  of 
any  given  sect. 

87. — What  relation,  if  any,  exists  between  obliques  drawn 
from  the  same  point  and  having  equal  projections? — unequal 
projections? 

88.— What  relation,  if  any,  exists  between  the  projections  of 
equal  obliques  drawn  from  the  same  point? — from  different 
points  in  the  same  perpendicular?— of  unequal  obliques  drawn 
from  the  same  point? — from  different  points  in  the  same  perpen. 
dicular? 

89. — What  kind  of  angle  does  the  shortest  sect  from  a  given 
point  to  a  given  line  make  with  the  line? 

90.— What  is  the  locus  of  a  point  equidistant  from  the  extrem- 
ities of  a  given  sect? 

9i. — Find  a  point  equidistant  from  three  given  independent 
points. 

92. — Show  how  to  find  the  center  of  a  circle  which  will  go 
through  the  three  vertices  of  a  given  trigon.  Such  a  circle  is 
said  to  be  circumscribed  about  the  trigon  through  whose  verti- 
ces it  passes,  and  its  center  is  called  the  circum=center  of  the 
trigon. 

93.— If  the  middle  point  of  the  base  of  an  isosceles  trigon  be 
joined  to  the  middle  point  of  one  of  the  legs,  what  relation,  if 
any,  exists  between  the  length  of  the  resulting  sect  and  the 
length  of  the  leg  of  the  trigon  ?  Try  three  or  four  different  tri- 
gons  before  coming  to  any  definite  conclusion. 

94.— By  the  aid  of  your  conclusion  in  q.  no.  93  devise  some 
method  of  erecting  a  perpendicular  when  its  foot  is  at  or  near  the 


LINEAR  RELATIONS.  39 

end  of  a  given  line. 

95. — Devise  some  means  of  dropping  a  perpendicular  from  a 
point  without  a  given  line,  when  the  point  lies  so  that  the  foot  of 
the  perpendicular  will  be  near  the  extremity  of  the  line. 

96. — Devise  some  means  of  drawing  the  bisector  of  a  given 
angle.    How  many  different  bisectors  may  the  same  angle  have? 

126. — By  the  distance  of  a  point  from  any 
line  (straight  or  curved)  is  meant  the  length  of  the 
shortest  sect  that  can  be  drawn  from  the  point  to  the 
line.  In  practical  drawing  we  frequently  find  the  dis- 
tance of  a  point  from  a  line  by  trial,  striking  arcs  of 
various  radii  about  the  point  as  center  until  we  find  the 
least  radius  whose  arc  will  reach  the  given  line.  This 
radius  then  gives  the  required  distance.  By  the  dis- 
tance of  a  point  from  a  sect  is  meant  the  distance  of  the 
point  from  the  seat  of  the  sect. 

Q.  and  E.  97. — What  relation,  if  any,  exists  between  the  dis- 
tances from  the  sides  of  an  angle  of  any  point  on  the  bisector  of 
the  angle? — of  any  point  between  the  sides  of  the  angle  and  not 
on  the  bisector? 

98. — What  is  the  locus  of  a  point  equidistant  from  any  two 
intersecting  lines.? — twice  as  far  from  one  as  from  the  other? — 
three  times  ? — n  times? 

99. — Devise  a  method  of  finding  a  point  within  a  trigon  which 
shall  be  equidistant  from  the  three  sides  of  the  trigon;  or  else 
show  that  there  is  no  such  point. 

100.— Within  a  given  trigon  draw  a  circle  which  shall  touch 
each  of  the  sides.  Such  a  circle  is  said  to  be  inscribed  in  the 
trigon  and  its  center  is  called  the  in-center  of  the  trigon. 

loi.— Draw  several  trigons  in  one  of  which  a  —  b ;  another, 
a  =  7.b ;  a  —  :^b  ;  a  =  ^b ;  a  =  $b ;  etc., —and  in  each  draw  the 
bisector  to  the  side  c.  What  general  relation,  if  any,  exists 
between  the  ratio  between  the  segments  of  the  third  side  and  the 
ratio  between  the  sides  to  which  those  segments  are  adjacent? 


40  LINEAR  RELATIONS. 

I02. — What  relation,  if  any,  exists  among  tlie  three  bisectors 
of  a  trigon  ? 

103.— Draw  the  three  altitudes  of  each  of  four  different  trigons, 
— one  of  which  is  scalene  and  acute,  one  isosceles  and  acute, 
one  right,  and  one  obtuse.  What  relation,  if  any,  is  there 
among  the  three  altitudes  of  each  of  these? 

104. — Is  it  possible  to  have  the  three  altitudes  of  a  trigon  with- 
in the  trigon? — two  only? — one  only? — none? 

105. — Draw  the  three  altitudes  of  a  trigon  in  which  the  sides 
are  proportional  to  two,  three,  and  four,  and  find  the  relation,  if 
any,  existing  among  the  lengths  of  these  altitudes. 

106. — Same  with  sides  proportional  to  three,  four,  and  five ; — 
to  four,  five,  and  six. 

107. — What  general  conclusion  seems  probably  true,  concern- 
ing the  ratio  between  any  two  sides  of  a  trigon  and  the  ratio 
between  the  altitudes  drawn  to  them? 

108. — In  a  scalene  trigon,  which  is  the  longest  altitude? — 
which  the  shortest? 

109.— Same  for  bisectors. 

1 10. — Same  for  medians. 

III.— What  relation  does  the  length  of  the  median  to  the 
hypotenuse  of  a  right  trigon  bear  to  the  length  of  the  hypote- 
nuse? 

112. — In  case  the  bisector,  the  median,  and  the  altitude  to  the 
same  side  of  a  trigon  are  all  unequal,  which  is  the  longest,  and 
which  the  shortest? 

113. — Which  two  may  be  equal  and  yet  differ  from  the  third? 

127. — Among  the  most  useful  of  a  draughtsman's 
instruments  are  his  triangles,  which  are  thin,  flat 
pieces  of  wood,  metal,  vulcanite,  or  other  hard  and  dur- 
able substance,  whose  outlines  are  those  of  trigons,  or 
triangles.  Each  has  usually  one  right  angle,  and  the 
acute  angles  are  of  such  sizes  as  are  most  frequently 
needed,— usually,    30'    and    6o%   ,and    45'    and    45°. 


LINEAR  RELATIONS.  41 

Throughout  the  rest  of  this  manual,  when  the  word 
triangle  is  used  a  draughtsman's  triangle  is  to  be  under- 
stood, unless  some  other  meaning  is  clearly  indicated. 

Q.  and  E.  114.— Devise  some  method  ot  testing  the  (supposed) 
right  angle  of  a  right  triangle,  using  a  ruler  or  another  triangle 
in  connection  with  the  one  to  be  tested.    See  art.  103,  page  31. 

115. — Devise  a  method  of  drawing  through  a  given  point  a 
line  perpendicular  to  a  given  line,  using  a  right  triangle  and  a 
ruler  or  another  triangle. 

116.— With  a  triangle  whose  angles  are  30°,  6o%  and  90°,  and 
another  whose  angles  are  45°,  45°,  and  90°  devise  means  of  con- 
structing angles  of  15%  75°,  and  105°. 

117.— Erect  three  or  more  perpendiculars  at  points  taken  at 
equal  intervals  on  any  straight  line  x,  constructing  the  perpen- 
diculars on  the  same  side  of  x,  and  call  these  perpendiculars, 
taken  in  order,  a,  d,  c,  etc.  Take  a  point,  A,  on  ^  at  a  conven- 
ient distance  from  x,  and  another,  B,  on  /^  at  a  trifle  less  distance 
from  X.  Through  A  and  B  draw  an  indefinite  straight  line, 
locating  C  where  this  line  crosses  c,  D  where  it  crosses  d,  etc. 
What  relation  exists  among  the  distances  of  A,  B,  C,  D,  etc., 
from  X  ? 

118. — Same  as  No.  117,  except  that  the  point  A  and  the  second 
point,  to  be  taken  on  a  perpendicular  at  a  considerable  distance 
from  a,  say  the  point  M  on  the  perpendicular  m,  are  to  be  taken 
equally  distant  from  x  and  on  the  same  side  of  it,  and  B,  C,  D, 
etc.,  are  to  be  located  where  the  lines  d,  c,  d,  etc.,  are  crossed  by 
the  line  through  A  and  M.  What  relation  exists  among  the  dis- 
tances of  the  points  A,  B,  C,  D,  etc.,  from  x? 

ii9.~If  a  straight  line  has  two  of  its  points  |  ""qually^  (  ^'^* 
tant  from  another  straight  line  lying  in  the  same  plane  (the  two 
points  spoken  of  being  on  the  same  side  of  the  second  line)  can 
the  lines  be  sufficiently  prolonged  to  cause  them  to  meet.'* 

120. — If  the  sides  of  one  angle  are  perpendicular  to  those  of 
another,  what  relation  exists  between  the  convex  angles  con- 
cerned? 

121.— If  the  sides  of  one  trigon  are  respectively  perpendicular 


42  LINEAR  RELATIONS. 

to  those  of  another  what  relation  exists  among  the  angles  of  the 
trigons  ? 

128. — Two  points  are  said  to  be  symmetric  about, 

,  ^        (  third  point  as  center  )     , 
or  with  respect  to,  a  i       ^  •    i-  •       r  when  this 

(  certain  line  as  axis     ) 

j  third   point  1  •    ^i^    j  mid-point  ) 

(        Hne        )  I  perpendicular    bisector  ) 

sect  joining  the  two  points.     This  i         .  Ms  called 

the  i         .       [  of    symmetry,    or,   more    briefly,   the 

(  sym-center  )  ^ 

i  .       h  for  the  two  points. 

(     sym=axis     )  ^ 

129 — Two  figures  are  symmetric  about  a  certain 

(  point  as  center  }     ,       .  ...  ,         . 

1     ,.  .       \  when  for  every  point  in  one  there  is 

(     line  as  axis     3  ^ 

a  symmetric  point  in  the  other  about  the  ]        .     [  afore- 
said. 

130. — A  single  figure  is  symmetric  about  a  certain 

\  point  as  center  |     ,       .,         ,,..,,. 

^     ,.  .       c  when  it  can  be  divided  into  two  parts 

(     line  as  axis     )  ^ 

which  are  symmetric  about  that  i  ^ ,.  .       [ 

(     line  as  axis.    ) 

IT      xu       V,  X  •      u     ^  (  a  center  i 

131. — For  the  phrase,  symmetric  about  i  .     r 

(  an  axis  ) 

,    „  ,       ,  (  sym=centric 

we  shall  use  the  shorter  one,  i 

(    sym=axic 

Q.  and  E.  122. — Is  a  scalene  trigon  sym-centric?    If  so,  about 
what  point? 

123. — Is  it  sym-axic?    If  so,  about  what  line.? 
124. — Same  for  isosceles  trigon. 
125. — Same  for  equilateral  trigon. 


."•! 


LINEAR  RELATIONS.  43 

126.— Same  for  acute  trigon, — right  trigon,— obtuse  trigon, — 
isogonic  trigon. 

127.-— Is  any  sect  sym-centric  ?    If  so,  about  what  point? 

128. — Is  any  sect  sym-axic?    If  so,  about  what  line? 

129. — Is  there  any  angle  which  is  sym-centric?  If  so,  what 
angle  and  about  what  point? 

130. — Are  there  any  open  lines  which  are  sym-centric  ?  If  so, 
draw  three  or  four  such,  curved  if  possible. 

131.— At  what  point  of  the  sect  joining  the  extremities  of  a 
sym-centric  curve  is  it  crossed  by  the  curve  if  at  all?  Need  it  be 
crossed  by  the  curve  if  the  curve  is  sym-centric?  In  case  it  is 
crossed,  what  relation  between  the  point  of  intersection  and  the 
sym-center? 

132. — Is  any  angle  sym-axic?    If  so,  about  what  line? 

133. — Are  there  any  open  lines  which  are  sym-axic?  If  so 
draw  three  or  four,  curved  if  possible. 

134. — If  the  extremities  of  one  sect  are  sym-centric  with  those 
of  another,  are  the  sects  sym-centric  ?    Are  their  seats? 

135. — Same  for  axic  symmetry. 

136. — Can  two  sym-centric  lines  meet? 

137. — Must  they  lie  in  the  same  plane? 

138. — Same  for  two  sym-axic  lines. 

139. — If  two  sym-axic  lines  meet,  where  does  their  point  of 
intersection  lie? 

140.— How  many  sym-centers  may  one  figure  have?— sym- 
axes? 

141.— Draw,  if  possible,  a  figure  having  three  or  more  sym- 
centers;— three  or  more  sym-axes. 

142.— If  a  sym-centric  figure  is  also  sym-axic,  is  or  is  not  its 
sym-center  necessarily  on  its  sym-axis? 

143. — If  a  sym-axic  figure  has  two  axes,  is  or  is  not  their  inter- 
section necessarily  a  sym-center  for  the  figure? 

144 — If  a  sym-axic  figure  has  three  or  more  sym-axes,  are  or 
are  they  not  necessarily  concurrent? 

132. — Any  figure  is  said  to  revolve  (in  the  geomet- 


44  LINEAR  RELATIONS. 

ric  sense  of  the  word)  about  a  given  fixed  point  when 
it  moves  so  that  each  point  in  it  maintains  a  fixed  dis- 
tance from  that  given  point.  The  given  point  is  the 
center  of  revolution. 

133. — Any  figure  is  said  to  revolve  about  the  line 

through  any  two  fixed  points  when  it  revolves  about 
each  of  those  two  points  at  the  same  time.  The  line 
aforesaid  is  called  the  axis  of  revolution. 

134. — A  thing  revolves  through  a  full  revolution 

about  any  axis  when  it  revolves  about  that  axis  until  it 
reaches  its  original  position,  without  having  retraced  any 
portion  of  its  path.  It  revolves  through  a  given 
angle  when  a  perpendicular  from  any  point  of  it  to  the 
axis  sweeps  over  the  given  angle. 

135. — It  must  be  remembered  that  these  uses  of  the 
word  revolve  are  the  only  ones  which  it  has  in  geome- 
try. Outside  of  geometry  it  is  used  in  other  and  wider 
senses. 

Q.  and  E.  145. — If  one  figure  sym-axic  with  another  be 
revolved  through  half  a  revolution  about  the  sym-axis,  what 
relation  will  it  then  bear  to  the  second  figure? 

146.  — If  one  plane  figure  sym-centric  with  another  in  the  same 
plane  be  revolved  through  half  a  revolution  (keeping  always  in 
the  plane)  about  the  sym-center,  what  relation  will  it  then  bear 
to  the  second  figure? 

147. — If  one  of  two  indefinite  perpendicular  lines  be  revolved 
about  the  other  as  axis,  what  will  its  path  be? 

148.— Can  a  line  meet  a  plane  in  such  a  way  that  it  shall  be 
perpendicular  to  every  line  lying  in  the  plane  and  drawn  through 
the  trace  of  the  given  line  upon  the  plane? 


LINEAR  RELATIONS.  45 

136. — The  centric  angle  of  any  arc  is  the  angle 
whose  vertex  is  at  the  center  of  the  arc,  whose  sides 
pass  through  the  extremities  of  the  arc,  and  which  is 
swept  across  by  the  arc.  The  arc  is  said  to  subtend 
its  centric  angle  and  to  be  intercepted  by  it. 

137. — The  chord  to  a  given  radius  of  any  angle 

is  the  chord  of  the  arc  intercepted  by  the  angle  and 

struck  with  the  given  radius  from  the  vertex  of  the 

angle  as  center.     When  no  length  of  radius  is  specified, 

the  radius  is  understood  to  be  of  unit  length. 

Q.  and  E.  149. — If  two  angles  are  equal,  what  relation  exists 
between  their  chords  to  the  same  radius?— what  if  they  are 
explementary? 

150. — If  two  angles  have  equal  chords  to  the  same  radius, 
what  relation  exists  between  the  angles?  What  if  the  chord  of 
one  is  larger  than  the  chord  of  the  other? 

151.— Devise  a  method  of  constructing  an  angle  which  shall 
have  a  given  side  and  a  given  vertex,  and  which  shall  be  equal 
to  a  given  angle. 

152. — By  the  method  called  for  in  ex.  151,  find  the  sum  of  the 
angles  of  a  scalene  trigon  ; — of  an  isosceles  trigon  ; — of  an  equi- 
lateral trigon. 

153.— Same  fot  trigons  with  sides  twice  as  large  as  in  those 
just  used  ; — thrice. 

154. — What  general  statement,  if  any,  seems  to  hold  true  in 
the  results  of  the  nine  exercises  just  called  for  in  ex.  152  and  153? 

155. — How  many  right  angles  may  any  trigon  have ? — obtuse? 
—concave  ? 

156. — If  in  two  trigons  ai  =  «2,  h  =  h,  and  71  =  72,  what  rela- 
tion exists  between  the  trigons  ? 

157.— What,  if  ai  =  ai,  (3i  =  ^2,  and  71  =  72? 

158. — What,  if  ai  =  «2,  h  —  hj  and  /^i  =  /?2? 

159. — Does  the  relation  between  the  lengths  of  a  and  d  in  ex. 


46  LINEAR  RELATIONS. 

158  have  anything  to  do  with  the  relation  between  the  trigons?— 
Does  the  size  of  /3? 

i6o.~If  b\  =  d'2,  A  =  1^2,  and  71  =  72,  what  relation  exists 
between  the  trigons  ? 

161.— What  if  ai  —  ^2,  /?i  =  /?2,  and  }i  =  72? 

162. — Can  there  be  two  trigons  in  which  ^i  =  ag,  /3i  =  /?2,  and 
7i  >  72? 

163. — The  sides  and  angles  of  any  trigon  are  customarily 
called  the  parts  of  the  trigon.  What  three  parts  of  one  trigon 
must  be  respectively  equal  to  three  parts  of  another,  in  order  that 
the  two  trigons  may  be  equal  or  opposite? 

164. — If  in  two  trigons  «i  =  ^2,  h  =  h,  but  71  >  72,  what  rela- 
tion exists  between  ci  and  C2  ?  Try  several  different  pairs  of  tri- 
gons before  announcing  any  general  conclusion. 

165. — If  ai  —  ai,  b\  =  bi,  but  c\  >  C2,  what  relation  exists 
between  71  and  72? — what  between  fti  and  /?2? — what  between  ^i 
and  ^2?  Try  several  different  pairs  before  announcing  any  gen- 
eral conclusion. 

138. — When  one  line  intersects  another  it  is  said  to 
traverse  that  other  at  the  point  of  intersection  and  is 
called   a  traverser   with   respect  to  that  other  line. 


When  a  traverser  traverses  two  lines  anywhere  else 
than  at  their  point  of  intersection  eight  angles  are 
formed,  which,  on  account  of  the  frequency  with  which 
the  case  occurs,  are  divided  into  groups,  and  the  groups 


LINEAR  RELATIONS.  47 

named.  Thus  if  AB  and  CF  are  the  traversees  {i.  e., 
the  things  traversed, )  and  GH  is  the  traverser,  crossing 
AB  at  Q  and  CF  at  R,  the  four  angles  between  the 
traversees,  AQR,  BQR,  CRQ,  and  FRQ,  are  called 
inner  angles;  the  other  four  are  outer  angles.  Any 
two  angles  lying  on  opposite  sides  of  the  traverser,  and 
not  adjacent,  are  called  alternate.  The  eight  angles 
are  divided  into  pairs  of  corresponding  angles  in 
these  three  ways: — any  pair  of  alternate  inner  angles; 
any  pair  of  alternate  outer  angles;  any  pair  of  non-adja- 
cent angles  lying  on  the  same  side  of  the  traverser  and 
being,  one  of  them  inner,  and  the  other  outer.  Either 
of  a  pair  of  corresponding  angles  is  said  to  be  the  corre= 
spondent  of  the  other. 

Q.  and  E.  166.— If,  when  two  lines  are  traversed  by  a  third, 
some  one  angle  equals  its  correspondent,  what  relation,  if  any, 
exists  between  any  other  angle  and  its  correspondent? 

139. — Two  lines  are  said  to  be  parallel  when  they 

lie  in  the  same  plane  and  have  such  positions  that  they 

can    never    meet,  no  matter  how  far  produced.     Two 

sects  are  parallel  when  their  seats  are  parallel.     The 

sign  for  ''parallel"  is  ||. 

Q.  and  E.  167.— Through  one  point  how  many  lines  can  be 
drawn  parallel  to  a  given  line? 

168.— If  each  of  two  given  lines  is  parallel  to  a  third  given 
line,  are  or  are  not  the  first  two  parallel  to  each  other? 

169.— If  a  straight  line  lying  in  the  plane  of  two  parallels  cross 
one  of  them,  will  or  will  it  not  cross  the  other  ? 

170.— At  least  how  many  of  the  seats  of  the  sides  of  a  trigon 
will  be  crossed  by  any  line  lying  in  their  plane? 


48  LINEAR  RELATIONS. 

171.— If  two  lines  lying  in  the  same  plane  be  crossed  by  a 
traverser,  and  their  positions  be  such  that  corresponding  angles 
are  equal,  on  which  side  of  the  traverser  will  the  traversees  meet? 

172.— If  they  meet  on  one  side,  must  they  not  also  meet  on  the 
other  side? 

173.— Of  what  kind  are  the  two  lines  mentioned  in  q.  171,— 
parallel,  perpendicular,  or  oblique? 

174.— If  two  parallel  lines  be  crossed  by  a  traverser,  what  rela- 
tion in  size  exists  between  corresponding  angles? 

175.— If  a  line  is  perpendicular  to  one  of  two  parallels,  what 
relation  does  it  bear  to  the  other? 

176.— If  one  pair  of  parallels  intersect  another  pair  of  parallels, 
what  relation  exists  between  the  segments  of  one  pair  intercept- 
ed by  the  other  pair? 

177.— What  may  be  said  of  the  distances  of  the  various  points 
of  a  given  line  from  another  line  to  which  the  first  is  parallel  ? 

178.— What  is  the  locus  of  a  point  at  a  given  distance  from  a 
given  line? 

179.— If  two  parallels  cut  a  circle  what  relation  exists  between 
the  chords  of  the  arcs  intercepted  by  the  parallels? 

180. — Devise  as  many  methods  as  you  can  for  drawing 
through  a  given  point  a  line  parallel  to  a  given  line. 

181.— If  three  parallels  make  equal  intercepts  on  one  traverser, 
what  relation  is  there  between  the  intercepts  they  make  on  any 
other  traverser  ? 

182.— What  relation  exists  between  any  side  of  a  trigon  and 
the  line  joining  the  middle  points  of  the  other  two  sides?  See 
q.  181,  also  q.  93. 

183.— If  from  any  point  whose  distances  from  two  intersecting 

"ines  are  in  a  given  ratio,  — '  sects  be  drawn  parallel  to  each 

until  they  meet  the  other  of  the  two  given  lines,  what  is  the  ratio 
between  these  sects,  and  what  is  that  between  the  distances  of 
their  extremities  in  the  given  lines  from  the  point  of  intersection 
of  the  given  lines? 

184. — Devise  an  easier  method  than  that  developed  in  q.  98, 
for  constructing  the  locus  of  a  point  whose  distances  from  two 


LINEAR  RELATIONS.  49 

intersecting  lines  shall  be  in  a  given  ratio. 

185.— Find  the  locus  of  a  point  equi-distant  from  two  non-par- 
allel lines  whose  intersection  is  off  the  drawing. 

186.— Devise  a  method  for  drawing  through  a  given  point  a 
traverser  which  shall  cross  two  non-parallel  lines  so  as  to  make 
the  inner  angles  on  the  same  side  of  it  equal. 

187.— Devise  a  method  by  which  to  divide  a  given  line  into 
any  desired  number  of  equal  parts. 

r88.— If  the  sides  of  one  angle  are  parallel  to  those  of  another, 
what  relation  is  there  between  the  convex  angles  concerned  ? 

189. — If  the  sides  of  one  trigon  are  parallel  to  those  of  another, 
what  relation  exists  among  the  angles  of  the  trigons? 

140. — If  a  collection  of  traversers  be  drawn  across 
a  collection  of  parallels,  the  segments  of  the  traver- 
sers intercepted  between  any  pair  of  parallels  are  called 
corresponding  segments. 

Q.  and  E.  190. — Draw  a  series  of  parallels,  and  call  them  m, 
71,  o,  p,  etc.;  then  draw  any  pair  of  traversers,  A  and  4.  The 
segment  of  h  which  is  between  m  and  7i  call  b\,  that  between  n 
and  o  call  c\,  etc.;  corresponding  segments  of  4  call  ^2,  ^2,  ^2,  etc. 
If  the  ratio  of  bi  to  b'i  equals  1%,  what  is  the  ratio  of  ci  to  C2, — of 
d\  to  ^2, — of  any  pair  of  corresponding  segments  of  A  and  /2?    If 

-;—  =  3  what  are  the  values  of  the  ratios  —— .  —7-.  etc.?     What 

6*2  ^2  «2 

when  y-  =  4X  ?    What  when  the  ratio  -^—  =  r  {r  meaning 
any  number)  ? 

191.— Complete  the  following  proposition:— "If  three  or  more 
parallels  are  crossed  by  two  traversers,  the  ratio  between  any 
two  corresponding  segments that  between  any  oth- 
er two,  taken  in  the  same  order." 

192. — When  these  traversers  are  concurrent  lines,  call  the 
point  of  concurrence  C,  and  the  segments  of  tu,  n,  etc.  between 
/i  and  fi  denote  by  ?7h,<> ,  ;zi,2 ,  etc., — those  between  /2  and  4  by 
"h.^ ,  ?zj.3 .  etc.     What  relation,  if  any,  exists  between  the  ratios 


50  LINEAR  RELATIONS. 

» 

—- ^»  —~^  etc.,  and  the  ratio  between  the  distances  of  m  and 

;/  from  C  ? 

193. — Complete  the  following  proposition: — "The  ratio  between 
the  segments  of  two  parallels  intercepted  by  two  concurrent  trav- 
ersers   the  ratio  between  the  distances  of  these  par- 
allels from  the  center  of  concurrence." 

194. — Calling  the  point  where  A  crosses  in.  Mi,  where  it  cross- 
es n,  Ni,  etc.,  what  relation,  if  any,  exists  between  the  ratios 

^1,2       ??^2,3        ^  J  ^u         ^.         CMi      CM2        ^    ^ 

— — »  — — »  etc.,  and  the  ratios  -r^^^  Tmm"'  etc.? 

^1,2       ^2,3  ClNi      CN2 

195. — From  the  principles  developed  in  the  preceding  exercis- 
es, deduce  a  method  for  finding  the  fourth  proportional  to  three 
given  sects;  /.  e.,  having  given  a,  b,  and  c,  find  a  sect  d  so  that 
a   _    c 
~T~~~d- 

196.— Deduce  a  method  of  finding  the  third  proportional  to 
two  given  sects ;   i.  e.,  having  given  a  and  b,  find  c  so  that 
a   _    b 
T~~c" 

197.— Deduce  a  method  of  dividing  a  given  sect  into  parts 
which  shall  be  proportional  to  given  sects. 

141. — The  relation  developed  in  example  192  leads  to 
a  very  useful  and  convenient  method  for  drawing 
through  a  given  point  a  line  which  if  sufficiently  pro- 
duced would  pass  through  the  point  of  concurrence  of 
two  lines,  this  point  of  concurrence  being  off  the  draw- 
ing. Suppose  the  two  lines  are  b  and  Cy  and  the  given 
point  is  K.  Call  the  center  of  concurrence  O.  Draw 
through  K  any  convenient  traverser,  /,  which  shall 
cross  b  at  Bj  and  c  at  Cj,  and  draw  another  traverser,  q, 
which  shall  be  parallel  to/,  and  at  as  considerable  dis- 
tance from  it  as  is  convenient.  Suppose  q  crosses  b  at  Bo 
and  c  at  Q.     Draw  B.^C,.     Draw  through  K  a  line  par- 


LINEAR  RELATIONS. 


51 


allel  to  b  meeting  BaQ  at  L,   and  draw  from  L  a  line 
parallel  to  c,  meeting  ^  at  M.     A  line  through  K  and  M, 


c.^ 


if  sufficiently   prolonged   will   pass  through   O.     The 
student  may  give  the  reasons.  ^ 

142. — In  drawing,  very  frequent  use  is  made  of  the 
relation  between  corresponding  angles  -when  two  paral- 
lels are  crossed  by  a  traverser.  One  of  the  most  com- 
mon instruments  whose  uses  depend  wholly  or  partially 
on  the  relation  just  mentioned  is  the  T  square  (sym- 
bol, To.)  It  consists  of  a  long,  thin,  straight-edged 
blade,  fastened  (or  capable  of  being  fastened )  rigidly  to 
a  **head,"  which  is  a  thicker  and  usually  a  much  shorter 
block  than  the  blade;  one  face  of  this  head  is  intended 
to  be  plane,   and  in  using  the  instrument,  this  face  is 


*  See  Eagles's  Const.  Geom.  of  Plane  Curves,  page  6,  prob.  4. 


52  LINEAR  RELATIONS. 

slid  along  the  straight  edge  of  the  drawing-board.  The 
appropriateness  of  the  name  appears  when  the  resem- 
blance of  the  instrument  to  the  letter  T  is  noticed.  If 
the  edge  of  the  drawing-board  along  which  the  Tn  head 
is  slid  be  truly  straight,  and  lines  be  drawn  along  the 
same  edge  of  the  blade  when  the  straight  edge  of  the 
head  is  put  in  contact  with  the  straight  edge  of  the 
board  at  different  places,  these  lines  will  be  parallel. 
(Why.?)  Other  very  common  instruments  used  fre- 
quently in  drawing  parallels  are  the  triangles,  men- 
tioned in  Art.  127,  page  40.  They  are  used  in  pairs  or 
else  one  is  used  singly  in  connection  with  a  ruler.  The 
principle  upon  which  depends  their  use  in  drawing  par- 
allels is  the  same  as  that  upon  which  depends  the  use 
of  the  Ta  for  like  purpose. 

Q.  and  E.  198. — With  two  triangles  or  with  a  triangle  and  a 
ruler,  devise  a  method  of  drawing  through  a  given  point  a  line 
parallel  to  a  given  line. 

199. — Devise  a  method  of  dropping  a  perpendicular  from  a 
given  point  to  a  given  line  when  the  distance  of  the  point  from 
the  line  exceeds  the  length  of  any  side  of  the  right  triangles  at 
hand,  and  is  also  larger  than  the  largest  spread  of  the  compass 
legs. 

143. — Any  plane  figure  bounded  by  sects  is  called  a 
polygon;  the  bounding  sects  are  its  sides,  their  extrem- 
ities the  corners  or  vertices,  and  the  interior  angles  at 
the  vertices  are  the  angles  of  the  polygon. 

144. — The  sum  of  the  sides  of  a  polygon  is  called  its 
perimeter. 

145. — Any  sect  joining  any  two  vertices  not  pertain- 


LINEAR  RELATIONS.  53 

ing  to  the  same  side  is  called  a  diagonal. 

146.— A  polygon  is  convex  when  all  its  angles  are 
convex;  otherwise,  it  is  concave. 

147. — A  concave  angle  of  a  polygon  is  sometimes 
called  a  re=entrant  angle. 

148. — If  any  two  sides  of  a  polygon  intersect,  it  is 
;alled  a  crossed  polygon;  otherwise  it  is  non=crossed. 

Polygons  are  to  be  understood  to  be  non-crossed,  unless 
the  context  clearly  indicates  that  the  most  general 
meaning  of  the  word  is  intended. 

149. — With  respect  to  the  number  of  their  angles, 
polygons  are  classified  as  trigons,  tetragons,  penta- 
gons, hexagons,  heptagons,  or  septagons,  octa- 
gons, nonagons,  decagons,  undecagons,  dodeca- 
gons, etc.,  according  as  they  have  thre^,  four,  five,  six, 
seven,  eight,  nine,  ten,  eleven,  twelve,  etc.,  angles. 
Those  of  twenty  angles  are  called  icosagons.  Those 
whose  number  of  angles  is  represented  by  the  algebraic 
number  n  we  shall  call  enagons;  /.  ^.,  instead  of  the 
phrase  "polygon  of  n  sides"  we  shall  use  the  word 
"enagon. " 

150. — Any  polygon  whose  angles  are  all  equal  is 
called  an  isogon;  one  whose  sides  are  all  equal  is  called 
an  equilateral.  One  both  isogenic  and  equilateral  is 
said  to  be  regular. 

151. — The  ends  of  the  links  of  a  chain  are  its  verti- 
ces; the  angles  at  its  vertices  are  its  angles. 


54  LINEAR  RELATIONS. 

1 52. — A  chain  is  convex  when,  of  every  four  consec- 
utive vertices,  the  sect  joining  the  first  to  the  third  is 
crossed  by  the  sect  joining  the  second  to  the  fourth; 
otherwise  it  is  concave. 

153. — Any  chain  is  isogonic  when  it  is  convex  and 

all  its  convex  angles  are  equal.     It  is  equilateral  when 

its  links  are  equal.     When  both  isogonic  and  equilateral 

it  is  regular. 

Q.  and  E.  200. — What  is  the  sum  of  the  angles  of  a  pentagon? 
-of  a  hexagon?— of  a  heptagon?— of  an  octagon?— of  an  enagon? 

201.  — Compute  the  size  of  an  angle  in  an  isogonic  trigon,— tet- 
ragon, — pentagon, — enagon. 

^^202. — Draw  an  isogonic  trigon,—tetragon,— hexagon,— octa- 
gon,—dodecagon. 

203. — What  regular  polygon  has  each  angle  equal  to  13/7  times 
a  right  angle?— 15/9  ?— i"/i5? 

204. — How  many  re-entrant  angles  may  a  trigon  have?— a  tet- 
ragon?—a  pentagon?— an  enagon? 

205. — Draw  an  equilateral  trigon,— tetragon,— pentagon,— hex- 
agon. 

206. — Can  an  equilateral  trigon  be  concave?— an  equilateral 
tetragon?— an  equilateral  pentagon? 

207. — How  many  convex  angles  must  there  be  in  an  equilat- 
eral tetragon  ?— pentagon  ?— hexagon  ?— heptagon  ?— enagon  ? 

208.— What  is  the  least  number  of  convex  angles  which  it  is 
possible  to  have  in  any  polygon  ? 

154. — A  line  is  tangent  to  a  circle  when  it  just 
touches  the  circle  without  crossing  it.  A  tangent  line 
thus  has  no  points  within  the  circle  to  which  it  is 
tangent. 

155. — A  sect  is  tangent  to  a  circle  when  it  has  a 


LINEAR  RELATIONS.  55 

point  in  common  with  the  circle  and  its  seat  is  tangent. 

156. — Any    polygon   or    chain   is    circumscribed 

about  a  circle  when  all  its  sides  or  links  are  tangent  to 
the  circle.  It  is  inscribed  in  a  circle  when  all  its  ver- 
tices are  points  on  the  circle. 

,       (  inscribed  in  ) 

157. — If  any  figure  be  i     .  •.    j    t.     ^  [•  a  cir- 

•^  (  circumscribed  about  ) 

,     ^,       .    ,     .     (  circumscribed  about  |  ^,     ^    , 
cle,  the  circle  is  i  .       .^    .  •  c  the  first-men- 

(         inscribed  in  ) 

tioned  figure. 

o      T-u         ^       r  ^v.      •    1    S         inscribed  in        ) 
1 58. — The  center  of  the  circle  i    .  •-■,,.     ^  r 

(  circumscribed  about  ) 

any  figure  is  called  the  ]    .  4.       [  oi  the  figure. 

^     ^  (  circumcenter  )  *=" 

Q.  and  E.  209.— Draw  the  perpendicular  bisectors  of  the  links 
of  a  regular  three-linked  chain.  What  relation,  if  any,  exists 
among  them? 

210.  — Is  a  regular  chain  inscriptible ;  2. «?.,  capable  of  being 
inscribed  in  a  circle? 

211.  — If  so,  show  how  to  find  its  circiim-center;  if  not,  show 
why  the  circle  through  any  three  consecutive  vertices  will  not 
also  pass  through  the  fourth. 

212.— Draw  the  bisectors  of  the  angles  of  a  regular  four-linked 
chain.    What  relation  ^  if  any,  exists  among  them  ? 
213.— May  a  circle  be  inscribed  in  any  regular  chain? 

214.  — If  so,  show  how  to  find  the  in-center  of  any  such  chain  ; 
if  not,  show  why  the  circle  touching  any  three  consecutive  links 
will  not  also  touch  the  fourth. 

215.  — If  a  circle  may  be  circumscribed  about  any  regular  chain^ 
and  another  may  be  inscribed  in  such  chain,  what  relation  exists 
between  their  centers? 

216. -Same  as  ex.  209  to  ex.  215,  for  regular  polygons. 
217.— Is  an  inscribed  equilateral  chain  regular  or  not? 


*  56  LINEAR  RELATIONS. 

218.— Same  for  isogenic  inscribed  chain. 
219.— Is  a  circumscribed  equilateral  chain  regular  or  not? 
220.— Same  for  circumscribed  isogonic  chain. 
221. — Same  as  q.  217  to  q.  220,  for  polygons. 
222. — A*re  any  tetragons  inscriptible? 

223.— Are  any  capable  of  being  circumscribed  about  a  circle.? 
224. — If  some  tetragons  are  inscriptible,  and  others  are  not, 
state  the  determining  conditions  so  far  as  you  can. 

225.— Same  for  those  capable  of  being  circumscribed  about  a 
circle,  if  there  are  any  such. 

226. — Are  any  regular  polygons  sym-  \  ^^Jil}^  [  ?  • 

227. — If  some  are,  are  all  ? 

228.— If  some  are  and  others  are  not,  state  the  determining 
conditions  for  each  sort  of  symmetry. 

229. — If  there  are  any  sym-centric  regular  polygons,  what  rela- 
tion exists  between  the  sym-center  and  the  in-center? 

230. — How  many  sym-axes,  if  any,  has  a  regular  enagon? 

231.— Same  as  q.  226  to  q.  230  for  regular  chains. 

159. — A  perpendicular  let  fall  from  the  in=center  of 
any  regular  polygon  or  chain  upon  any  side  or  link  is 
called  an  apothem. 

160. — A  sect  drawn  from  the  circumcenter  of  any 
regular  polygon  or  chain  to  any  vertex  is  called  a 
radius. 

Q.  and  E.  232.— What  relation,  if  any,  exists  among  the  vari- 
ous apothems  of  a  regular  polygon  or  chain  ?— radii  ? 

233. — At  what  point  does  an  apothem  meet  the  side  or  link  to 
which  it  is  drawn? 

234.— What  relation  exists  between  the  radii  of  a  regular  poly- 
gon or  chain  and  the  angles  to  whose  vertices  they  are  drawn? 

161. — When  we  speak  of  the  radius  or  tJie  apothem  of 


LINEAR  RELATIONS.  57 

a  regular  polygon  or  chain,  we  mean  the  leiigth  of  a 
radius  or  of  an  apothem  as  the  case  may  be. 

162. — The  angle  included  by  any  two  radii  drawn  to 
consecutive  vertices  of  a  regular  polygon  or  of  a  regular 
chain  is  the  centric  angle  of  such  polygon  or  chain. 

Q.  and  E.  235. — What  relation,  if  any,  exists  among  the  vari- 
ous centric  angles  of  any  regular  polygon  or  regular  chain? 

236. — What  is  the  size  of  a  centric  angle  of  a  regular  trigon? — 
tetragon  ? — pentagon  ? — enagon  ? 

163. — Tetragons  are  sometimes  called  quadrilater= 

als,  also  quadrangles;  the  reasons  for  these  names 

are  readily  apparent.     Tetragons  are  usually  classified 

with  respect  to  their  sides. 

164. — Any  tetragon  in  which  no  two  sides  are  par- 
allel is  called  a  trapezium. 

165. — Any  one  in  which  two  sides  are  parallel  and 
unequal  in  length  is  called  a  trapezoid.  The  parallel 
sides  of  a  trapezoid  are  called  its  bases ;  the  other  two 
its  legs.  When  the  two  legs  are  equal  the  trapezoid  is 
isosceles ;  otherwise,  it  is  scalene.  The  line  joining 
the  middle  points  of  the  legs  of  a  trapezoid  is  called  the 
median  of  the  trapezoid.  The  perpendicular  distance 
between  the  bases  of  a  trapezoid  is  the  altitude  of  it. 

166.— Any  tetragon  in  which  two  sides  are  parallel 
and  equal  in  length,  is  a  parallelogram.     The  two 

parallel  sides,  upon  one  of  which  the  parallelogram  is 
supposec  to  rest,  are  the  bases.  The  other  two  sides 
are  the  legs      Wheii  the  two  legs  are  equal,  the  paraU 


58  LINEAR  RELATIONS. 

lelogram  is  isosceles ;  otherwise,  it  is  scalene. 

167. — Parallelograms  whose  angles  are  right  angles 
are  called  rectangles,  others  are  called  oblique  paraU 
lelograms.  When  the  word  parallelogram  is  used 
hereafter,  an  oblique  non-equilateral  parallelogram,  or 
rhomboid  as  it  is  frequently  called,  is  to  be  understood 
unless  the  context  indicates  that  the  general  meaning  is 
intended. 

168. — Equilateral  parallelograms  are  called  rhom- 
buses when  oblique,  squares  when  rectangular. 

Q.  and  E.  237.— Draw  a  trapezium,— a  trapezoid, —an  isosce- 
les trapezoid, — a  crossed  trapezoid.  Are  isosceles  crossed  trape- 
zoids possible?— are  scalene.? 

238. — Draw  a  parallelogram, — an  isosceles  parallelogram,— a 
crossed  parallelogram, — a  scalene  parallelogram.  Are  isosceles 
non-crossed  parallelograms  possible? — scalene?  Are  isosceles 
crossed?— scalene? 

239.— Draw  a  rhombus,— a  crossed  rhombus. 

240.— Draw  a  rectangle, — a  square. 

241. — Draw  the  diagonals  of  each  of  the  figures  above  called 
for.    How  many  in  each  ? 

242. — If  two  consecutive  sides  of  a  trapezium  are  equal,  and 
the  other  two  are  also  equal,  what  relation,  if  any,  exists  between 
the  diagonals? 

243.— In  such  a  figure,  in  what  way  do  the  diagonals  divide 
the  angles  from  whose  vertices  they  are  drawn  ? 

244. — Can  such  a  trapezium  be  a  crossed  trapezium  ? 

245.— How  many  sym-axes,  if  any,  has  such  a  trapezium?  If 
there  be  any,  show  how  to  draw  them  (pr  it,  if  there  be  but  one). 

246.— How  many  sym-centers,  if  any,  has  such  a  trapezium? 

247. — What  relation,  if  any,  exists  between  the  angles  at  a 
base  of  an  isosceles  trapezoid?— scalene? 


LINEAR  RELATIONS.  59 

248.— What  relation,  if  any,  exists  between  the  angles  adjacent 
to  a  leg  of  a  trapezoid  ? 

249. — How  many  sym-axes  has  an  isosceles  trapezoid,  if  any? 
— sym-centers? 

250. — Same  for  scalene  trapezoid. 

251.— Are  your  conclusions  in  q.  247  to  q.  250  true  for  crossed 
trapezoids? 

252. — What  relation  exists  between  the  length  of  a  median 
and  the  lengths  of  the  two  bases  of  a  trapezoid? 

253. — How  far  would  the  median  of  a  trapezoid  have  to  be 
produced  to  meet  the  longer  base,  produced  if  necessary? 

254.— At  which  of  its  points  is  a  diagonal  of  a  trapezoid 
crossed  by  the  median? 

255.— Are  the  two  diagonals  and  the  median  of  a  trapezoid 
concurrent  lines? 

256.— If  not,  on  which  side  of  the  median  is  the  intersection  of 
the  diagonals  ? 

257. — What  relation  is  there  between  the  lengths  of  the  two 
diagonals  of  a  scalene  trapezoid ?— of  an  isosceles? 

258.— What  relation  exists  between  any  two  consecutive 
angles  of  a  parallelogram? — any  two  alternate  angles? 

259. — What  relation  exists  between  any  pair  of  alternate  sides 
in  a  non-crossed  parallelogram  ?  The  answer  to  this  shows  the 
reason  for  the  name. 

260. — Does  it  make  any  difference  which  two  opposite  sides  of 
a  parallelogram  are  taken  for  the  bases? 

261.— What  relation  does  the  point  of  intersection  of  the  two 
diagonals  bear  to  each? 

262.— How  do  the  diagonals  divide  the  angles  from  whose 
vertices  they  are  drawn  ? 

263.— What  relation  exists  between  the  two  trigons  into  which 
a  diagonal  of  a  parallelogram  divides  it? 

264.— What  relation  exists  between  the  lengths  of  the  diago- 
nals of  a  parallelogram? 

265.— Are  the  diagonals  ever  perpendicular? 

266._How  manv  sym-axes  has  a  parallelogram?— sym-centers? 


6o  LINEAR  RELATIONS. 

267.— Same  as  q.  258  to  q.  266  for  rhombus. 

268. — Same  as  q.  258  to  q.  266  for  rectangle. 

269.— Same  as  q.  259  to  q.  266  for  square. 

270.— Representing  the  consecutive  sides  of  a  tetragon  by 
a,  b,  c,  and  d,  the  angle  between  any  two,  a  and  b  for  instance, 
by  l.ab,  and  the  diagonal  joining  the  vertices  of  lab  and  /.ca 
by  g\,  the  other  by  ^2,  state  the  conditions  necessary  to  the 
equality  of  two  trapeziums;— of  two  trapezoids;— of  two  parallel- 
ograms;—of  two  rhombuses;— of  two  rectangles;— of  two  squares. 

169. — Two  rectilinear  figures  are  similar  when 

the  angles  of  one  are  respectively  equal  to  those  of  the 

other  and  arranged  in  the  same  order,  while  the  sides 

of  one  are  proportional  to  the  sides  of  the  other  and 

arranged  in  the  same  order  with  respect  to  the  angles. 

170.— Any  ]     .J     [of  one  and  its  ]        ^^^^.       ,  \ 
(   side  )  (  proportional  ) 

in  the  other  of  two  similar  figures  are  said  to  be  homol= 
ogous  when  their  relative  positions  in  the  figures  are 
the  same.  The  vertices  of  any  two  homologous  angles 
are  homologous.  If  two  things  are  homologous,  either 
is  the  homologue  of  the  other. 

171. — Two  similar  figures  are  similarly   placed 

when  the  lines  through  any  three  vertices  of  one  and 
their  respective  homologues  are  concurrent  (the  center 
of  concurrence  not  being  between  any  two  homologous 
vertices, )  and  the  distances  of  the  three  vertices  of  one 
figure  from  the  center  of  concurrence  are  proportional 
to  those  of  their  homologues. 

172. — Any  point  of  one  of  two  similar  figures  is  homol- 
ogous to  a  certain  point  of  the  other  if  when  the  two  fig- 


LINEAR  RELATIONS.  6i 

ures  are  similarly  placed  the  sects  joining  the  point  in  one 
to  any  two  of  its  vertices  are  parallel  to  the  sects  joining 
the  point  in  the  other  to  the  homologous  vertices. 

173. — A  sect  in  one  of  two  similar  figures  is  homolo- 
gous to  one  in  the  other  if  the  extremities  of  the  first 
are  homologous  to  those  of  the  second. 

174. — Two  chains  are  similar  when  upon  drawing 
their  chords  {i.  ^.,the  lines  connecting  their  extremities) 
two  similar  polygons  result,  of  which  the  chords  are 
homologous  sides.  Any  two  things  of  the  chains  arc 
homologous  if  they  are  homologous  things  in  the  similar 
polygons  which  result  when  the  chords  are  drawn. 

175. — The  ratio  of  similitude,  or  the  linear  ratio, 

between  two  similar  figures  is  the  ratio  of  any  side  of 
the  first  to  its  homologue  of  the  second. 

176. — Shape  being  dependent  upon  the  relative  posi- 
tions of  the  different  parts  of  a  figure  with  respect  to 
each  other,  and  these  relative  positions  being  the  same 
in  any  two  similar  figures,  it  is  frequently  said  that 
similar  figures  have  the  same  shape. 

Q.  and  E.  Make  at  least  six  different  comparisons  in  each 
case  before  announcing  any  conclusion  concerning  the  relations 
below  inquired  about. 

271.— If  two  unequal  trigons  are  mutually  isogonic  (/.  e.,  have 
the  angles  of  one  respectively  equal  to  those  of  the  other),  are 
they  similar  or  not? 

272.— Same  for  two  tetragons. 

273.— Same  for  two  pentagons. 

274.— Same  for  any  two  polygons  of  more  than  three  sides. 


62  LINEAR  KELATIONS. 

275. — Same  for  two  trigons  having  two  angles  of  one  respect- 
ively equal  to  two  of  the  other.  If  so,  show  why ;  if  not,  give 
the  conditions  which  determine  in  which  of  the  two  trigons  the 
third  angle  is  the  larger. 

276.— If  two  trigons  have  one  angle  of  one  equal  to  one  angle 
of  the  other,  and  the  two  sides  including  that  angle  in  the  first 
proportional  to  the  two  sides  including  its  equal  in  the  second, 
are  or  are  not  the  trigons  similar? 

277.— If  two  trigons  have  the  sides  of  one  respectively  propor- 
tional to  those  of  the  other,  are  the  trigons  similar  or  not  similar? 

278. — Are  two  tetragons  similar  or  not,  under  like  conditions? 
— two  pentagons  ? — any  two  polygons  of  more  than  three  sides  ? 

279. — If  the  altitude  upon  the  hypotenuse  of  a  right  trigon  be 
drawn,  into  what  sort  of  trigons  does  it  divide  the  original  trigon? 

280.— What  relation  does  each  bear  to  the  other  and  to  the 
original  trigon? 

28!.— Of  the  two  segments  into  which  this  altitude  divides  the 
hypotenuse,  what  relation  does  the  ratio  of  one  segment  to  the 
altitude  bear  to  the  ratio  of  the  altitude  to  the  other  segment? 

282. — Knowing  that  the  three  vertices  of  a  right  trigon  are 
equi-distant  from  the  middle  point  of  the  hypotenuse,  devise 
some  means  of  constructing  a  mean  proportional  between  two 
given  sects. 

283. — What  relation  does  the  ratio  between  the  perimeters  of 
two  similar  trigons  bear  to  their  ratio  of  similitude? 

284. — Same  for  homologous  altitudes. 

285. — Same  for  any  two  homologous  sects. 

286.— Same  as  q.  283  to  q.  285,  for  similar  polygons  of  any 
number  of  sides. 

287. — If  any  two  similar  polygons  be  divided  into  trigons  by 
drawing  homologous  diagonals,  what  relation  exists  between  the 
trigons  of  one  polygon  and  those  of  the  other  ? 

288. — What  relation  exists  between  the  arrangement  in  one  set 
and  that  in  the  other? 

'289. — If  any  two  polygons  be  made  up  of  similar  trigons,  those 
in  one  being  respectively  similar  to  those  in  the  other  with  the 
same  ratio  of  similitude  throughout,  and  occupying  the  same  rel- 


LINEAR  RELATIONS.  63 

ative  positions,  what  relation  exists  between  the  polygons? 

290. — What  relation  does  the  map  of  a  plane  surface  bear  to 
the  surface  of  which  it  is  a  map? 

291.— If  the  distance  between  two  places  be  24K  miles,  what 
will  be  the  number  of  inches  between  their  representatives  on  a 
map  drawn  to  a  scale  of  i  to  10000? 

177. — If  through  any  center  of  concurrence,  O,  an 

indefinite  number  of  rays  be  drawn,  a,  b,  c,  etc.,  and  on 

each  ray  a  pair  of  points  be  taken,  Ai  and  Ao  on  a,  B, 

and  B2  on  b,  etc.  so  that  the  ratios  7^'  7^'  etc.  are  all 

OA2     Vjt)2 

equal,  and  so  that  if  O  is  between  one  pair  it  shall  be 

between  every  other  pair,  the  figure  composed  of  the 

points  Ai,  Bi,  Q,  etc.  is  said  to  be  homothetic  to  that 

composed  of  the  points  A2,  B2,  C.,  etc.;  the  center,  O,  is 

called  the  homothetic  center,  or  the  center  of  honio= 

thesy;   the   rays  a,  b,  c,  etc.  are  called  homothetic 

OA 
rays;  and  the  ratio  ^^  is  called  the  homothetic  ratio 

or  the  ratio  of  homothesy.     The  figures  are  directly 

homothetic  when  the  center  is  not  between  any  pair 
of  corresponding  or  homologous  points  such  as  Aj  and 
A2.  When  it  is  between  such  a  pair,  the  figures  are 
inversely  homothetic.  The  relation  existing  be- 
tween two  figures  by  virtue  of  which  they  are  homo- 
thetic is  called  homothesy. 

Q.  and  E.  292.— If  one  of  two  homothetic  figures  is  a  straight 
line,  what  is  the  other  figure?  What  relation  does  it  bear  to  the 
first  in  size?— in  position? 

293.— If  one  of  two  homothetic  figures  is  a  polygon,  what  kind 
of  figure  is  the  other  ? 


64  LINEAR  RELATIONS. 

294.— How  is  it  placed  with  respect  to  the  first  when  the  two 
are  directly  homothetic? — inversely? 

295.— What  relation  does  the  ratio  of  any  side  of  the  first  to  its 
homologue  in  the  second  bear  to  the  homothetic  ratio? 

296.— Making  use  of  the  relations  existing  between  homothetic 
figures,  devise  a  method  of  drawing  a  polygon  similar  to  a  given 
polygon,  and  having  a  given  ratio  of  similitude  to  it,  using  a  cen- 
ter of  homothesy. 

297.— If  F'  and  F"  are  two  figures,  both  homothetic  to  a  third 
figure  F,  but  about  different  centers,  O"  and  O'  respectively,  are 
or  are  not  F'  and  F"  homothetic  to  each  other?  Try  half  a  doz- 
en different  cases  before  announcing  a  conclusion. 

298.— If  so,  what  relation  does  their  center  of  homothesy,  O, 
bear  to  the  other  two  centers,  O"  and  O'  ? 

299.— If  not,  show  what  is  lacking  to  their  being  homothetic  to 
each  other. 

PROBLEMS. 


178. — A  problem  is  a  proposition  or  statement  of 
something  proposed  or  commanded  to  be  done. 

179. — The  solution  of  a  problem  is  the  actual 
accomplishment  of  the  thing  commanded,  or  else  the 
method  of  the  accomplishment. 

180. — The  discussion  of  a  problem  is  the  examina- 
tion of  the  reasons  underlying  the  solution  and  of  the 
cases  in  which  more  than  one  solution  may  be  possible 
or  in  which  no  solution  may  be  possible. 

181. — In  solving  problems,  loci  should  be  used 
whenever  possible.  Occasionally  a  solution  is  indi- 
cated through  a  consideration   of   the   relations   which 


LINEAR  RELATIONS.  65 

would  hold  between  given  things  and  required  things  if 
the  solution  had  actually  been  obtained.  The  use  of 
one  or  more  of  these  relations  in  connection  with  known, 
or  given,  things  may  lead  to  the  desired  solution. 

Give  full  solutions  and  discussions  of  the  following 
problems. 

Find  a  point  X  which  shall  be 

300.— Equidistant  from  three  given  points,  A,  B,  and  C. 

301. — Equidistant  from  two  given  points,  A  and  B,  and  at  a 
given  distance  from  a  third  point,  C. 

302. — At  given  distances  from  two  given  points,  A  and  B. 

303.— Equidistant  from  two  given  points,  A  and  B,  and  on  a 
given  line  m. 

304.— Equidistant  from  two  given  points  A  and  B,  and  at  a  giv- 
en distance  from  a  given  line  vi. 

305.— Equidistant  from  two  given  points  A  and  B,  and  also 
from  two  given  lines  in  and  ;^— (i),  parallel;  (ii),  intersecting. 

306. — At  a  given  distance  from  a  given  point  A,  and  on  a  given 
line  771, 

307. — At  a  given  distance  from  a  given  point  A,  and  at  a  given 
distance  from  a  given  line  771. 

308.— At  a  given  distance  from  a  given  point  A,  and  equidis- 
tant from  two  given  lines,  7n  and  /«,—  (i),  parallel ;  (ii),  intersect- 
ing. 

309. — Equidistant  from  three  given  lines, — (i),  three  lines  par- 
allel ;  (ii),  two  lines  parallel ;  (iii),  no  two  lines  parallel ;  (iv), 
three  lines  concurrent. 

310.— On  a  given  line  and  equidistant  from  two  others,— (i), 
parallel ;  (ii),  intersecting. 

311. — At  a  given  distance  from  one  given  line  and  equidistant 
from  two  others,— (i),  parallel ;  (ii),  intersecting. 

312. — At  given  distances  from  two  given  lines,— (i),  parallel: 
(ii),  intersecting. 

313.— Draw  the  complete  locus  of  a  point  which  shall  be  two 


66  LINEAR  RELATIONS. 

times  as  far  from  one  of  two  given  lines  as  from  the  other,  (i) 
lines  parallel,  (ii)  lines  intersecting ;— three  times  ;— four  times; 
—five  times.    See  ex.  183  and  184,  page  48. 

314.— Two  times  as  far  from  a  and  three  times  as  far  from  b  as 
from  ^,— (i)  a,  b,  and  c  parallel,— (ii)  a  and  b  parallel,— (iii)  no 
two  lines  parallel,— (iv)  a,  b,  and  c  concurrent. 

3i5.^In  one  side  of  a  trigon  and  equidistant  from  the  other 
two ;— twice  as  far  from  the  second  as  from  the  third ;— three 
times. 

316.— Find  a  point  which  shall  divide  one  side  of  a  trigon  into 
segments  proportional  to  the  other  two  sides.  See  ex.  loi, 
page  39. 

317.— Draw  a  line  across  a  trigon  so  that  it  shall  be  parallel  to 
the  base  and  the  segment  included  between  the  legs  (or  the  legs 
produced)  shall  have  a  given  length  ;— 

318.— shall  be  equal  to  the  sum  of  the  segments  of  the  legs 
included  between  it  and  the  base. 

319.— Draw  a  sect  which  shall  be  parallel  to  a  given  line  a, 
have  its  extremities  in  two  other  given  lines  b  and  c,  and  have  a 
given  length,— (i),  a,  b,  and  c  parallel;  (ii),  only  b  and  c  parallel; 
(iii),  only  a  and  b  parallel;  (iv),  no  two  lines  parallel,  and  lines 
non-concurrent;  (v),  a,  b,  and  c  concurrent. 

320.— Find  the  path  of  the  middle  point  of  a  sect  whose  ends 
move  in  right  lines  that  are  perpendicular  to  each  other.  See  q. 
Ill,  page  40. 

321. — If  P  and  Q  are  two  points  on  the  same  side  of  a  given 
line  X,  draw  the  shortest  two-linked  chain  which  shall  connect 
P  and  Q  and  touch  x.  What  relation  exists  between  the  acute 
angles  its  links  make  with  xl 

322. — Given  any  point  P  between  the  sides  of  an  angle,  to 
draw  a  sect  terminating  in  those  sides  and  bisected  at  P.  See  q. 
182,  page  48. 

In  the  exercises  following,  a,  b,  and  c  stand  for  the  sides  of  a 
trigon, — a,  /?,  and  y  for  the  angles  opposite  them  respectively, — 
^a,  K,  and  K  for  the  altitudes  to  them  respectively, — m^,  m^,,  m^ 
the  respective  medians,— </a,  d\,,  d^,  the  respective  bisectors,—/, 
the  perimeter, — r,  the  radius  of  the  inscribed  circle, — 7?,  the 
radius  of  the  circumscribed  circle. 


LINEAR  RELATIONS.  67 

Construct  the  trigon  which  shall  have  given  values  for 

323.  a,  b,  and  c,  or  a,  b,  and/. 

324.  a,  b,  and  m^. 

325.  a,  b,  and  k^. 

326.  a,  b,  and  a. 

327.  «,  <^,  and  y. 

328.  a,  »Za,  and  viYi. 

329.  a,  5^2a,  and  //a. 

330.  ^,  ?«a,  and  p. 

331.  ^,  ?;2b,  and  m^, 

332.  «,  ??2b,  and  h^. 

333.  «,  ;?Zb,  and  7. 

334.  «,  -^a,  and  ^. 

335.  a,  ^b,  and  /3. 

336.  <at,  <3?'b,  and  7. 

337.  «,  a,  and  /3. 

338.  «,  i3,  and  y. 

339.  ^,  ;?,  and/. 

340.  ??2a»  ^b,  and  ;?2c ; — consider  the  trigon  formed  by  drawing 
through  the  extremities  of  one  median  of  a  trigon  lines  parallel 
to  the  other  two. 

341.  m^,  ^b,  and  h^. 

342.  m^,  m\>,  and  he ;— consider  the  relation  between  the  alti- 
tude upon  any  side  of  a  trigon  and  the  distance  from  that  side  to 
the  point  of  concurrence  of  two  medians. 


343- 

;;^a,  /^a,  and  !^. 

344. 

m^,  >^b,  and  a. 

345- 

Wa,  0",  and  /^. 

346. 

m^,  l3,  and  y. 

347- 

/u,  /zb,  and  /. 

348- 

^a,  ^a,  and  a. 

349- 

ha,  ^a,  and  ,/3, 

350. 

/^a,  ^b,  and  /I 

351- 

^a,  a,  and  /3,  or  //»,  ^^,  and  /. 

68  LINEAR  RELATIONS. 


352 

-^a, /?,  and/. 

353 

<fa,  «,  and  /3,  or 

354 

^a,  /3,  and  7. 

355 

a,  /3,  and/. 

356 

a,  b-c,  and  /?. 

357 

a,  b—c,  and  7. 

358 

a,  ^— r,  and/. 

359 

a, /^-^,  and/?— 7. 

360 

^,  a-|-^,  and  a—^^ 

361 

a,  a-h/?,  and  /?H-7. 

362 

a,  a+/?,  and  /3-7. 

363 

a,  a-p,  and  /3-|-7. 

364 

^-<:,  a,  and  /?. 

365 

^— <:,  a,  and  7. 

366 

b—c,  /3,  and  7. 

367 

^-r,  a,  ^  (or  7),  and  a±/3,  (or  /3±7). 

368 

7;2a,  (or  ?;?b  or  z;2c),  ?^?a  ±  ^?b,  and  ?;2b  ±  ?^^c. 

In  the  isosceles  trigons  below  called  for,  b  represents  a  leg^  r 

the  be 

ise. 

Construct  the  isosceles  trigon  which  shall  have  sjiven  values 

for 

369 

b  and  <:  or  (^  and/. 

370 

(^  and  ?;2b. 

371 

<^  and  ?;2c. 

372 

.    b  and  /? 

373 

.    b  and  7 

374 

;?2b  and  /zc> 

375 

.     ?;2b  and  /?. 

376 

.    myy  and  7. 

377 

.    m\y  and  ^^c- 

378 

.    /2b  and  (3. 

379 

/i^  and  7. 

380 

/^b  and  ^c- 

381 

.    ^band/?. 

LINEAR  RELATIONS.  69 


382. 

^b  and  y. 

383. 

/?andA 

384. 

7  and/. 

385. 

b—c  and  y. 

386. 

b-c  and  /?. 

387. 

b—c2iXidip 

388. 

c  and  h^. 

389. 

c  and  y. 

390. 

c  and  /5. 

391. 

c  and/. 

392. 

c  and  Wb 

In  the  right  trigons  below  called  for,  a  and  b  are  the  legs,  c  is 

the  hypotenuse. 

Construct  the  right  trigon  which  shall  have  given  values  for 

393. 

a  and  b. 

394. 

a  and  c. 

395- 

«and/. 

396- 

a  and  c-b. 

39/. 

a  ana  n^. 

398. 

a  and  nif,. 

399- 

a  and  m^,. 

400. 

a  and  z?2c. 

401. 

a  and  ^b. 

402. 

^  and  d^. 

403. 

a  and  a. 

404. 

^  and  /3. 

405. 

^  and  a-/3. 

406. 

c  and/. 

407. 

<r  and  ^— <a!. 

408. 

c  and  a. 

409. 

r  and  a— /S. 

410. 

/  and  a. 

411- 

<:— ^  and  a. 

412. 

c—a  and  /3. 

70  LINEAR  RELATIONS. 


413 

c—a  and  a—^. 

414. 

b-a  and  a. 

415. 

b—a  and  /^. 

416. 

he  and  m.^. 

417. 

he  and  m^. 

418. 

he  and  d^ 

419 

he  and  a. 

420. 

he  and  a—^. 

421. 

m^  and  ?; , 

422. 

??2a  and  c. 

423. 

m^  and  ■' 

424. 

d^  and  ( 

425. 

d^  and  /". 

426 

dc  and  c. 

427. 

dc  and  a~X 

Construct  the  isogenic  trigon  which  shall  have  a  given  value 
for 

428. 

a. 

429. 

P- 

430 

h. 

431- 

a-h. 

432. 

a+h. 

433- 

p-h 

434- 

p-^h. 

If  a,  b,  c,  and  d  are  consecutive  sides  of  a  tetragon,  p  is  the 
perimeter,  Aab  is  the  angle  between  the  sides  a  and  b,  gx  is  the 
diagonal  joining  the  vertex  of  lab  to  that  of  led,  and  ^2  the 
other  diagonal,  construct  the  trapezium  having  given  values  for 

435.     a,  b,  c,  d,  and^i, 

436. 

a,  b,  c,  d,  and  Z  ab. 

437 

a,  b,  cgu  and  ^2, 

438. 

«,  b,  c,£-i,  and  lab, 

439- 
440. 

a,  b,  c,  gx,  and  I  cd, 
a,  b,  c,  gi,  and  Z  da, 

LINEAR  RELATIONS.  71 

441.  a,  b,  c,  lab,  and  Abe,- 

442.  a,  b,  c,  Aab,  and  Acd, 

443.  a,  b,  c,  /_ab,  and  Ada, 

444.  a,  b,  c,  Acd,  and  Ada, 

445.  a,  b,p,gu3in^  Abe, 

446.  <3!,  b,  p,  g2,  and  Z  be, 

447.  <3!,  (^, /,  Za(^,  and  Abe, 

448.  a,  ^,^1,  Aab,  and  Z<^r:, 

449.  «,  b,gi,  Abe,  and  Z^^, 
450  a,  b,  g^,  A  be,  and  Z  <r^, 

451.  <2,  <^,  Aab,  Abe,  and  Aed, 

452.  «2,  r,^,  ^1,  and  Z(^<r, 

453.  a,  ^,/, ^2,  and  Aed, 

454.  a,e,gi,  Aab,  and  Abe, 

455.  a,e,g\,  Abe,  and  Aed, 

456.  ^,  <r,  ^2,  Z  (^r,  and  Z  ^^, 

457.  a,  e,  A  ab,  A  be,  and  Z  <:^, 

458.  a,p,gx,  Aab,  and  Ada, 

459.  a,p,  Aab,  Abe,  and  Aed, 

460.  a,  p,  A  ab,  A  ed,  and  Z  ^^, 

461.  a,gi,g2,  Aab,  and  Abe, 

462.  a,gi,gi,  Aab,  and  Ada, 

463.  ^, i^i,  Z<3:^,  Abe,  and  Aed. 

In  the  exercises  below  given  in  the  construction  of  trapezoids, 
a  and  e  are  the  two  bases,  and  a  >  e.  The  symbols  have  the 
meanings  assigned  in  the  exercises  in  the  construction  of  tetra- 
gons, so  far  as  they- correspond.  Of  the  other  symbols,  m  rep- 
resents the  median,  h  the  altitude.  Construct  the  trapezoid 
which  shall  have  given  values  for 


464. 

a,  b,  e,  and^i, 

465. 

a,  b,  e,  and  g2, 

466. 

a,  b,  e,  and  Aab, 

467. 

a,  b,  e,  and  h. 

468. 

a,  b,  d,  and  Aab, 

n  LINEAR  RELATIONS. 


469. 

a,  b,  d,  and  h. 

470. 

a,  b,  p,  and  ^2, 

471. 

a,  b,p,  and  Lab, 

472. 

a,  b,  p,  and  Z  da. 

473- 

a,  b,p,  and  h. 

474- 

a,  b,gu  and  ^2, 

475. 

a,  b,,  gu  and  m, 

476. 

a,  b,gx,  and  /Lab, 

477- 

a,  b,  gu  and  Z  da, 

478. 

a,  b,gi,  and  h. 

479- 

a,  b,  g2,  and  Z  da. 

480. 

a,  b,  go,  and  m. 

481. 

a,  b,   Z  ab,  and  Z  da. 

482. 

a,  b,   z  ab,  and  m, 

483. 

a,  b,   Z  da,  and  m, 

484. 

a,  b,  Lda,  and  h. 

485. 

a,  c,  d,  and^i, 

486. 

a,  c,  d,  and  Z  ab. 

487. 

a,  c,  gi,  and  Z  <a!<^, 

488. 

«,  <r,^i,  and  h. 

489. 

a,  c,   Z  ab,  and  Z  ^(35, 

490. 

a,  c,   Z  ^<^,  and  /z, 

491. 

a,p,gu  and/i. 

492. 

a,  p,   z  ab,  and  Z  ^-a;, 

493. 

«,/,  Zab,  and  //, 

494- 

<a!,  /,  m,  and  /z, 

495. 

^,i^i,^2,  and  Z«^, 

496. 

^^£'u£'2,  and  /z, 

497- 

«,i^i,  Z«/^,  and  Zda, 

498. 

a,gi,  Zab,  and  m, 

499. 

a,gu  Zab,  and  ^, 

500. 

a,gi,  Zda,  and  z?2, 

501. 

^,  Z  ^/^,  Z  ^<2,  and  m, 

LINEAR  RELATIONS.  73 

502.  «,  Z.ab^  /Lda,  and  ^, 

503.  a,  Aab,  m,  and  h, 

504.  b,  d,gu  and  Z^<^, 

505.  <^,  ^,  ^1,  and /^, 

506.  b,  d,  Aab,  and  m, 

507.  /^,  d,  m,  and  >^, 

508.  b,p,gx,  and  Z^^, 

509.  <^,/,^i,  and/^, 

510.  b^p,  Z«^,  and  Z^<«, 

511.  b,p,  lab,  and  ?;2, 

512.  (^,/,  Z^(2,  and/z, 

513.  ^,/,  /;^,  and  h, 
5U-  b,gx,g2,  and  Z^3, 
515-  <^,  ^1,  .^2,  and /z, 

516.  ^,^1,  Z<2^,  and  /.da, 

517-  <^»^i»  Z«/^,  and?;2, 

518.  b,  g\,  m,  and  /^, 

519.  b,  Lab,  /.da,  and  m, 

520.  ^,  Z  ^<2, 7?2,  and  //, 
521.*  p,  gu  g2,  2ind  A, 

522.  /,  Z  ab,  Z  ^<^,  and  ^, 

523.  p,  /Lab,  Z da,  and /t, 

524.  p,  Lab,  m,  and  ^, 

525.  g\,  Lab,  /da,  and  h, 

526.  ^1,  Z«/^,  ^/,  and  ^, 

527.  /ab,  /da,  m,  and  -^. 

Construct  the  rhomboid  which  shall  have  given  values  for 

528.  a,  b,  and  g\, 

529.  a,  b,  and  /ab, 

530.  a,  b,  and  //, 

531.  «,/,  and^i, 

532.  a,p,  and  Z«^, 

533.  a,p,andh. 


74  LINEAR  RELATIONS. 

534-  «,^i,  and^2, 

535-  ^^^u  and  Aab, 

536.  a,  gu  and  h, 

537.  a,  Aab,  and  ^, 

538.  a,  lab,  andg2—b, 

539.  ^,  Z«^,  and^i— <^, 

540.  A.^1,  and//, 
541-  ^1  £'1^  and  «;— /J*, 

542.  /,  Z  ^i^,  and  /z, 

543.  /,  Z«^,  and  a— ^, 

544.  /,  /^,  and  tz— (J*, 

545.  ^,  a—b,  andgi-a, 

546.  .^1,  ^2,  and  /^, 

547.  ^1,  Zfl!/5',  and  /i, 

548.  ^1,  //,  and  a—b, 

549.  Z^^, /z,  and^^i— «. 

Construct  the  rhombus  which  shall  have  given  values  for 

550.  a  and  Zab, 

551.  aandgi, 

552.  a  and  ^1—^2, 

553.  «  and  /i, 

554.  Z^-^andi^i, 

555.  Z«^  and  ^1—^2 

556.  Z<:?^and>^, 
■JS?-  ^iand^2, 
55S.  gi  and  /^. 

Construct  the  rectangle  which  shall  have  given  values  for 

559.  a  and  b, 

560.  «  and/, 

561.  aand^i, 

562.  ^  and  ^1— «. 

563.  /and^i, 

564.  /  and  a—b, 


LINEAR  RELATIONS.  75 


565.-^^1  and  a—b. 
Construct  the  square 
566.     a. 

which  shall  have  a  given 

value  for 

567. 
568. 

569. 

g, 
P+g, 

570. 

g—ci. 

571. 

p-g> 

A  polygon  is  said  to  be  inscribed  in  another  when 
its  vertices  are  points  on  the  sides  of  that  other.  Make 
use  of  the  principles  of  homothesy  in  solving  following 
problems. 

572. — Devise  a  method  by  which  to  inscribe  in  a  given  trigon 
one  similar  to  another  given  trigon,  so  that  a  designated  side  of 
the  inscribed  trigon  shall  be  parallel  to  a  given  line.  How  many 
solutions?  Does  the  number  of  solutions  depend  upon  the  char- 
acters of  the  given  trigons  ?    Give  a  complete  discussion. 

573. — Devise  a  method  by  which  to  inscribe  a  square  in  a  giv- 
en trigon.  How  many  solutions?  Does  the  character  of  the 
trigon  have  anything  to  do  with  the  number  of  solutions?  Do 
the  inscribed  squares  differ  in  size,  if  more  than  one  may  be 
inscribed? 

574. — Same  for  inscribing  rectangle,  and  similar  questions. 

575.— Same  for  inscribing  parallelogram,  with  similar  ques- 
tions. What  relation  must  exist  between  the  angles  of  the  tri- 
gon and  those  of  the  given  parallelogram,  in  order  that  there 
may  be  a  solution  ? 

576. — Same  for  inscribing  trapezoid,  with  similar  questions. 

577.— Same  for  inscribing  trapezium,  with  similar  questions. 

578.— Is  it  always  possible  to  draw  a  tetragon  which  shall  be 
similar  to  a  given  tetragon  and  have  its  vertices  on  three  given 
lines, — (i),  lines  all  parallel;  (ii),  only  two  lines  parallel;  (iii), 
non-concurrent,  and  no  two  parallel ;  (iv),  concurrent?  If  possi- 
ble in  some  of  these  cases  and  impossible  in  others,  distinguish 
those  in  which  it  is  possible  from  those  in  which  it  is  not,  and 


76  LINEAR  RELATIONS. 

give  reasons  for  the  distinction. 

579.— Same  for  three  given  lines  and  given  trigon,  with  the 
restriction  that  a  designated  side  of  resulting  trigon  shall  be  par- 
allel to  a  fourth  given  line  parallel  to  none  of  first  three. 

580. — Similar  to  q.  578,  but  for  pentagon  and  four  given  lines, 
— (i),  four  lines  parallel;  (ii),  only  three  lines  parallel;  (iii),  two 
lines  parallel  to  each  other,  and  other  two  lines  parallel  to  each 
other  but  not  to  first  two;  (iv),  only  two  lines  parallel,  no  three 
lines  concurrent;  (v),  two  lines  parallel,  three  lines  concurrent; 
(vi),  no  two  lines  parallel,  no  three  concurrent;  (vii),  no  two  par- 
allel, only  three  concurrent;  (viii),  four  lines  concurrent.  Make 
distinctions  between  possible  and  impossible  cases,  if  there  are 
such,  and  give  reasons. 


CHAPTER  II. 

AREAL  RELATIONS. 

182. — As  has  before  been  said,  things  are  measured 
only  by  comparing  them  with  others  of  the  same  char- 
acter. The  area  or  extent  of  any  surface  is  measured, 
then,  only  by  comparing  it  with  the  extent  of  some 
other  surface  whose  extent  is  taken  as  unit. 

183. — In  practice,  we  almost  always  take  for  unit  of 
area  that  of  a  square  whose  side  has  unit  length; 

this  is  done  because,  for  most  purposes,  such  a  unit  is 
the  most  convenient  one  that  could  be  employed,  as  will 
appear  later.  When  no  other  unit  of  area  is  expressed, 
the  unit  of  area  is  to  be  understood  to  be  that  of  a 
square  the  length  of  one  side  of  which  is  equal  to  the 
unit  of  length  used. 

184. — Although,  as  has  just  been  said,  areas  are  found 
by  comparison,  they  are  found  always  by  indirect 

comparison.  Our  direct  comparisons  are  always  of 
one  line  with  another,  and  from  the  results  of  these  we 
compute  the  area  in  any  case.  As  will  appear  later, 
whatever  our  direct  measurements  may  be,  they  reduce 
ultimately  to  finding  the  lengths  of  two  lines  perpendic- 
ular to  each  other.     These  two  lengths  are  called  the 


78  AREAL  RELATIONS. 

dimensions  {i.  e.,  measurements)  of  the  figure  to 
which  they  pertain.  It  is  true  that  in  some  figures  of 
special  shape,  ^.  ^.,  squares  and  circles,  one  measure- 
ment seems  sufficient,  but  in  reality  besides  the  one 
direct  measurement  we  need  to  know  the  additional  fact 
that  the  second  direct  measurement,  if  made,  would 
give  a  result  bearing  a  definite  known  relation  to  that  of 
the  first. 

185. — The  dimensions  of  a  surface  are  named  lengtli 
and  breadth.  If  they  are  unequal,  the  name  length  is 
to  be  applied  to  the  larger  unless  the  contrary  is  stated 
or  implied.  The  word  width  sometimes  takes  the 
place  of  the  word  breadth  in  geometric  usage. 

186.— When  we  speak  of  the  products  or  of  the 
quotients  of  sects,  limited  surfaces,  etc.,  or  of  lengths, 
breadths,  areas,  etc.,  we  mean  the  products  or  the 
quotients  of  their  respective  enumerators.  See  Art.  16, 
page  4. 

187. — The  dimensions  of  a  trigon  or  of  a  parallelogram 
are  the  length  of  its  base  and  that  of  its  altitude;  those 
of  a  trapezoid  are  the  length  of  its  median  and  that  of 
its  altitude.  Other  figures  than  those  of  these  three 
classes  have  their  areas  found  by  being  divided  into 
parts  coming  under  one  or  more  of  these  classes,  the 
sum  of  the  areas  of  which  gives  the  area  desired. 

188. — By  the  rectangle  upon  two  given  sects  is 

meant  a  rectangle  whose  base  and  altitude  are  equal  to 


AREAL  RELATIONS.  79 

the  given  sects  respectively.  By  tlie  rectangle  of  two 
given  sects  is  meant  the  product  of  those  two  sects. 
Similarly  for  the  square  upon  any  sect,  and  the 
square  ^/any  sect. 

Q.  and  E.  581. — What  is  the  ratio  between  the  areas  of  two 
rectangles  having  equal  altitudes,  if  their  bases  are  equal  ?— if 
the  base  of  the  first  is  two  times  that  of  the  second  ?— three 
times? — seven  and  a  half  times?— 263^  times?— ;^  times? 

582. — What  relation  exists  between  the  area  of  a  rectangle  and 
the  product  of  its  dimensions? 

583. — In  order  that  your  answer  to  the  preceding  question  may 
be  true,  is  it  necessary  that  both  dimensions  be  expressed  in  the 
same  unit,  or  not?  If  so,  what  relation  does  the  unit  of  area  in 
this  case  bear  to  the  unit  of  length? 

584.— Find  the  dimensions  of  ten  different  rectangles,  the  area 
of  each  of  which  shall  be  360  square  feet. 

585. — If  the  area  of  a  rectangle  and  one  dimension  of  it  be  giv- 
en, how  may  the  other  dimension  be  found? 

586.— Why  should  the  product  of  a  number  by  itself  be  called 
the  square  of  the  number? 

587.— Why  should  the  product  of  two  numbers  be  called  the 
rectangle  of  those  numbers? 

588.— Find  the  relation  existing  between  the  square  upon  the 
sum  of  two  sects  and  the  rectangle  and  squares  upon  those  two 
sects. 

589.— Same  for  the  square  upon  the  difference  between  two 
sects. 

590.— Find  the  relation  existing  between  the  rectangle  upon 
the  sum  of  two  sects  and  their  difference,  and  the  squares  upon 
those  two  sects. 

591.— Compare  the  results  of  the  last  three  exercises  with  the 
algebraic  formulas  for  {a-\-bf,  {a—bf^  and  (^+^)  {a—b). 

592. — Draw  a  rectangle  and  an  oblique  parallelogram,  having 
equal  bases  and  equal  altitudes.  What  relation  exists  between 
their  areas?  Which  is  the  larger?  Make  half  a  dozen  compari- 
sons before  announcing  any  definite  conclusion. 


8o  AREAL  RELATIONS. 

593.— How  may  the  area  of  a  parallelogram  be  found  when  its 
dimensions  are  known? 

594.— What  relation  exists  between  the  ratio  of  the  areas  of 
two  parallelograms  and  the  ratio  of  their  altitudes  when  their 
bases  are  equal? — the  ratio  of  their  bases  when  their  altitudes 
are  equal  ? 

595. — What  relation  exists  between  the  area  of  a  trigon  and 
that  of  a  parallelogram  having  two  sides  in  common  with  the  tri- 
gon?—between  the  altitudes  of  the  two  upon  their  common  base? 

596.— How  may  the  area  of  a  trigon  be  found  when  its  two 
dimensions  are  known  ?    Give  the  reason  for  your  answer. 

597.— If  one  side  of  a  trigon  is  twice  as  large  as  another,  what 
is  the  ratio  of  the  altitudes  upon  those  sides? — if  n  times? 

598. — In  a  scalene  trigon  can  two  altitudes  be  equal? — three? 

599.— In  any  trigon  in  which  no  two  altitudes  are  equal,  to 
which  side  does  the  longest  altitude  pertain? — the  shortest? 

600.— If  two  altitudes  of  a  trigon  are  equal  what  kind  of  a  tri- 
gon is  it? — what,  if  all  three  are  equal  ? 

601.— What  is  the  ratio  between  the  areas  of  two  trigons  hav- 
ing equal  bases  and  equal  altitudes  upon  those  bases? — equal 
bases  and  unequal  altitudes  upon  them?^-equal  altitudes  and 
unequal  bases?— the  same  base  and  their  vertices  on  a  line  par- 
allel to  the  base? — the  same  vertex  and  their  bases  upon  the 
same  line?    Give  your  reason  for  your  answer  in  each  case. 

602. — In  two  trigons,  if  ax  =  ^2,  and  n  =  72,  but  ^1  >  or  <  ^2, 
what  relation  exists  between  the  ratio  of  the  area  of  the  first  to 
that  of  the  second  and  the  ratio  of  di  to  <^2?— which  ratio  is  invar- 
iably the  larger? 

603. — Extend  your  conclusion  in  the  preceding  to  the  case 
where  71  =  72  but  ^1  >  or  <  a2,  and  ^1  >  or  <  ^2 ; — also  to  that 
in  which  71+72  =  180°,  and  ^i  >  or  <  a^  and  /^i  >  or  <  ^2. 

604. — If  two  equivalent  trigons  have  a  common  base  and  lie  on 
the  same  side  of  it,  what  relation  does  the  sect  joining  their  two 
vertices  bear  to  the  common  base?    Give  your  reason. 

605. — If  through  the  middle  point  of  one  leg  of  a  trapezoid  a 
sect  be  drawn  parallel  to  the  other  leg,  and  extended  until  it 
meets  the  longer  base  and  the  shorter  base  produced,  thus  cut- 


AREAL  RELATIONS.  8i 

ting  off  a  trigon  in  one  place  and  adding  one  in  another,  what 
kind  of  figure  will  result? 

606.— What  relation  does  its  base  bear  to  the  median  of 
the  trapezoid?— its  altitude  to  the  altitude  of  the  trapezoid? 
— its  area  to  the  area  of  the  trapezoid  ?  Give  your  reasons  for 
thinking  so. 

607.— How  may  the  area  of  a  trapezoid  be  found  when  its  two 
dimensions  are  known? — when  its  altitude  and  its  two  bases  are 
known?    Give  reasons. 

608.— If  all  the  radii  of  a  regular  polygon  be  drawn,  into  what 
sort  of  figures  is  the  polygon  divided  ? 

609.— How  may  the  area  of  a  regular  polygon  be  found  when 
its  apothem  and  its  perimeter  are  known? 

610. — How  may  the  area  of  a  polygon  circumscribed  about  a 
circle  be  found  when  the  perimeter  of  the  polygon  and  the  radius 
of  the  inscribed  circle  are  known? 

611.— If  the  two  diagonals  of  a  tetragon  are  perpendicular  to 
each  other,  how  may  the  area  of  the  figure  be  found  when  the 
lengths  of  these  two  diagonals  are  known? 

612.— Does  your  answer  to  the  preceding  question  hold  true  in 
the  case  af  a  re-entrant  tetragon  with  perpendicular  diagonals? 

189.— The  area  of  a  trapezium  or  of  any  irregular 
polygon  of  five  or  more  sides  may  be  found  by  drawing 
a  sufficient  number  of  diagonals  to  divide  the  polygon 
into  trigons,  and  taking  the  sum  of  the  areas  of  these 
trigons.  Usually  however  it  is  more  convenient  to 
draw  the  longest  diagonal  and  drop  perpendiculars  upon 
it  from  the  various  vertices,  thus  dividing  the  polygon 
into  trapezoids  and  right  trigons,  the  sum  of  whose  areas 
is  the  area  required. 

190. — Occasionally  it  is  desired  to  reduce  a  polygon 
to  an  equivalent  trigon  by  a  purely  graphical  method. 
The  method  followed  in  such  case  will  now  be  consid- 


82 


AREAL  RELATIONS. 


ered.  Suppose  A,  B,  C,  F,  G,  H,  etc.,  the  consecutive 
vertices  of  the  polygon  to  be  reduced  and  suppose  that 
AB  is  taken  as  the  sect  whose  seat  is  to  be  the  seat  of 
the  base  of  the  required  trigon.  From  B  draw  the  diag- 
onal to  the  second  consecutive  vertex  after  it,  F  in  this 
case,  thus  dividing  the  polygon  into  the  trigon  BCF 
and  the  polygon  ABFGH  etc.  Now  if  the  trigon  BCF 
can  be  replaced  by  an  equivalent  trigon  which  shall  still 
have  BF  for  one  side  but  shall  have  its  third  vertex,  Q, 
say,  in  AB  or  AB  produced  and  on  the  same  side  of  BF 
as  C  is,  it  is  evident  that  the  polygon  AQFGH  etc.,  will 
be  equivalent  to  the  original  and  will  have  one  less  side. 


By  a  consideration  of  his  reply  to  question  No.  604, 
page  80,  the  student  will  see  how  to  draw  the  sect  CQ 
so  that  the  desired  relation  between  the  trigons  BCF 
and  BQF  shall  be  obtained.  The  process  is  precisely 
the  same  when  the  angle  C  is  concave  as  when  it  is 
convex.  Evidently  the  same  sort  of  an  operation  may 
be  performed  on  the  polygon  AQFGH  etc.,  by  which  a 


AREAL  RELATIONS. 


83 


polygon  equivalent  to  the  original  and  having  two  less 
sides  will  be  obtained.  By  a  sufficient  number  of  repe- 
titions of  this  operation,  any  polygon  may  be  reduced 
to  an  equivalent  trigon. 

Q.  and  E.  613. — Reduce  a  convex  heptagon  to  an  equivalent 
trigon,  and  then  reduce  the  resulting  trigon  to  an  equivalent 
rectangle. 

614. — Reduce  a  hexagon  with  two  concave  angles  to  an  equiv- 
alent rectangle. 

191. — There  is  a  very  important  relation  existing 
among  the  three  sides  of  a  right  trigon  which  will  now 
be  considered.     Suppose   ABC   any   right  trigon,  the 


right  angle  having  its  vertex  at  C.  Prolong  CA  to  J 
making  AJ  equal  to  CB,  and  thus  having  CJ  equal  to 
the  sum  of  CA  and  CB.  Also  prolong  CB  to  F  making 
BF  equal  to  CA,  and  CF  equal  to  CJ.  Through  F  and 
J  draw  sects  parallel  to  CJ    and   CF  respectively,  to 


84  AREAL  RELATIONS. 

meet  at  a  point  which  call  H.  On  the  sect  FH  take  G, 
and  on  the  sect  HJ  take  I,  so  that  FG  and  HI  shall  each 
equal  CB.  Draw  the  three  sects  AI,  IG,  and  GB. 
Draw  a  sect  from  B  parallel  to  CJ  and  meeting  JH  at 
Q,  and  from  A  draw  a  sect  parallel  to  CF  meeting  BQ 
at  K,  and  from  G  draw  one  parallel  to  HJ  meeting  BQ 
at  L.     Then  trace  out  the  relations  called  for  below. 

Q.  and  E.  upon  preceding  article. 

615.— What  kind  of  figure  is  AIGB  and  what  relation  does  it 
bear  to  BA? 

616.— What  kind  of  figure  is  AJQK  and  what  relation  does  it 
bearto  JQ?— toBC? 

617.— What  kind  of  figure  is  QHGL  and  what  relation  does  it 
bear  to  HG?— to  tA? 

618.— What  relations  do  the  trigons  AJI,  IHG,  GFB,  GBL, 
and  AKB  bear  to  the  trigon  ABC? 

619.— What  relation  does  the  sum  of  the  two  tetragons  AKBC 
and  GFBL  bear  to  the  sum  of  the  four  trigons  AJI,  IHG,  GFB, 
andBCA? 

620.— If  from  the  whole  figure  there  be  subtracted  the  four 
trigons  just  mentioned,  there  remains  the  tetragon  AIGB,  and  if 
from  the  whole  figure  there  be  subtracted  the  two  tetragons  men- 
tioned in  q.  619,  there  remains  a  figure  which  is  the  sum  of  the 
two  tetragons  AJQK  and  QHGL.  What  relation  exists  between 
the  tetragon  AIGB  and  the  sum  of  the  tetragons  AJQK  and 
QHGL? 

621. — What  relation  exists  between  the  square  upon  the 
hypotenuse  of  a  right  trigon  and  the  sum  of  the  squares  upon  the 
two  legs  of  the  trigon  ? 

622.— If  the  lengths  of  two  of  the  sides  of  a  right  trigon  be 
known,  how  may  the  length  of  the  third  side  be  computed,— (i) 

two  legs  given,— (ii)  the  hypotenuse  and  one  leg  given? 

'623. — What  relation  exists  between  the  sum  of  the  squares  of 
two  sides  of  a  trigon  and  the  square  of  the  third  side,  if  the  first 


ARHAL  RELATIONS.  85 

two  sides  include  an  acute  angle? — a  right  angle? — an  obtuse 
angle?    See  q.  164,  page  46. 

624.— If  the  lengths  of  the  sides  of  a  trigon  are  known,  how 
may  it  be  determined  without  drawing  the  trigon,  whether  it  is 
acute  or  right  or  obtuse  ? 

625. — Determine  the  character  (with  respect  to  angles)  of  the 
trigon  whose  sides  are,— 6,  7,  9;  5,  12,  13;  2,  3,  4;  5,  6,  8;  12,  12, 
23;  6,  8,  10;  I,  2,  2K;  iH,  2,  2K;  16,  22,  and  23. 

192. — By  means  of  the  relation  connecting  the  lengths 
of  the  legs  of  a  right  trigon  and  that  of  the  hypotenuse, 
the  student  may  easily  deduce  a  formula  for  the  length 
of  the  altitude  upon  any  side  of  a  trigon  in  terms  of  the 
three  sides,  and  another  for  the  area  of  the  trigon  in 
terms  of  the  three  sides.  Thus,  let  a,  d,  and  c  be  the 
sides,  c  being  taken  as  base,  and  suppose  the  altitude 
4  drawn.  If  ^  >  or  <  <^,  suppose  a  >  d.  The  projec- 
tion of  a  upon  c  call  /,  and  that  of  d  upon  c  call  g.  Now, 
if  a  is  acute, p-\-q  =  c;  if  a  is  right, p  =  c  and  ^  =  o,  so 
that /-f^  still  =  c;  if  a  is  obtuse,  p—g  -c.  In  all  cases 
p'^h:  =  a\  and  (f^K  =  ^,  so  that  /-^  =  d-b". 

When  p^q  =  c,  p-q  =  -^'    ^^^^"^  =5t^' 

then     {p^q)  +  {p-q)  =  2p  =  ^+— —  =  -^ ' 

or         p  =  — ' 

When  p—q  =  c,  p-^q  = '  and  2/  =  c-\- 

—  —^ '  so  that  p  =  — ' as  before. 


86  AREAL  RELATIONS. 

Now     h^  =  d—f  =  i^^-p)  i^—p) 

~  2C  2C 

{a-^cf-b'    b'-{a-cY 

X 


2C  2C 

_  {a-\-c-{-b){a-\-c—b)     {b^a—c){b—a-\^c) 

~  2c  2C 

_  {a^b-\-c)  {a-i^b-\-c—2b)  {aJ^b-\-c—2c)  {aJ^b-\-c~2a) 

AC' 
Substitute  s  for  the  semi-perimeter,  i.  e.,  put 


^^^if, 


or  2s  —  aA-b-^c. 

2 
,2         (  2^ )  ( 2S--2b )  ( 2S—2c){2S~2a) 

n^  —  — 

4c 

_  i65(^— <^)  {s—b)  {s—c) 
AC' 

-  ^\s{s—a){s—b){s—c)\^       whence 

2 


4=  ~^s{s-a){s-b){s-c).  (0 

Putting  ^4^01*  the  area  of  the  trigon,  we  have,  since 


combining,  A  =  ^s{s-a)  (s-b)  (s-c) .  (2) 

On  account  of  their  great  practical  importance,  form- 


AREAL  RELATIONS.  ^7 

ula?  (i)  and  (2)  should  be  carefully  remembered. 

193. — By  means  of  the  relation  among  the  three  sides 
of  a  right  trigon,  a  method  of  constructing  a  square 
equivalent  to  a  given  rectangle  may  be  developed. 
Suppose  a  diudi  b  the  legs  of  a  right  trigon,  c  the  hypote- 
nuse, h  the  altitude  upon  Cy  p  the  projection  of  a  on  c, 
and  q  the  projection  of  b  on  c.  Then  we  have  c  =  p-\-q, 
and  ^  zr  /-|-^2+2/^ ;  but  ^  =  d'-Yb\  and  d  =  p^-yh' 
while  b"  =  q'-\-/i\  so  that  c'  =  /^Z^^+^+Z/^  =  /+/-}-2Z/l 
Then  2Z/^  =  2pq,  or  /{^  =  pq;  i-  e.,  the  square  upon  the 
altitude  drawn  to  the  hypotenuse  of  a  right  trigon  is 
equivalent  to  the  rectangle  upon  the  projections  of  the 
legs  on  the  hypotenuse.  Thus,  to  construct  a  square 
equivalent  to  a  given  rectangle,  lay  off  upon  a  straight 
line  two  adjacent  sects  respectively  equal  to  two  consec- 
utive sides  of  the  given  rectangle;  at  the  junction  point 
of  these  sects  erect  a  perpendicular  and  upon  it  take 
such  a  point  as  may  serve  as  the  vertex  of  a  right  angle 
whose  two  sides  shall  pass  through  the  outermost 
extremities  of  the  sects  before  mentioned.  This  point 
determines  such  a  portion  of  the  perpendicular  erected 
as  will  equal  a  side  of  the  required  square.  A  reference 
to  q.  Ill,  page  40,  will  suggest  a  means  of  determining 
the  desired  point  upon  the  perpendicular. 

194. — It  will  be  noticed  that  in  the  discussion  just  giv- 
en, since  pq  -  Z/^  h  is  a  mean  proportional  between  /  and 

q ;  for,  dividing  each  member  by  qh,  we  have  -^  =  — ; 


88  AREAL  RELATIONS. 

i.  e.y  in  any  right  trigon  the  projection  of  one  leg  on  the 
hypotenuse  is  to  the  altitude  upon  the  hypotenuse  as 
that  altitude  is  to  the  projection  of  the  other  leg  on  the 
hypotenuse.  The  method  of  finding  the  side  of  a  square 
which  shall  be  equivalent  to  a  given  rectangle  also 
enables  us  then  to  find  the  mean  proportional  between 
two  given  sects.     In  this  connection  it  is  to  be  particu- 


larly  noticed  that  if  h^  -  pq,  ^  -  ^^\   for   when   h^ 


9 
Jj  -    q 


p        h 
pq,-j-= — .     Multiply  both  sides  of  last  equation  by 

~-  and  there  will  result  —^  =  -4-x —  =  -^. 
k  h'        Jf'      q        q 

Q.  and  E.  626.— Which  has  the  larger  perimeter,  a  rectangle 
or  its  equivalent  square? 

627. — Reduce  a  given  concave  hexagon  to  an  equivalent 
square. 

628.— Devise  a  method  for  constructing  graphically  the  square 
roots  of  2,  3,  5,  6,  7,  8,  and  10,  assuming  any  convenient  sect  to 
represent  unity.  What  relation  exists  between  the  lines  you  get 
representing  the  square  roots  of  2  and  8? — what  among  unity, 

V2,  V3,  and  V6? 

195. — The  ratio  between  the  areas  of  two  figures  is 

called  their  areal  ratio. 

Q.  and  E.  629.— What  relation,  if  any,  exists  between  the  areal 
ratio  of  two  similar  trigons  and  their  ratio  of  similitude,  or  linear 
ratio  ? 

630. — Same  for  similar  polygons. 

631. — Supply  the  missing  words  (if  there  are  any)  in  the  fol- 
lowing propositions: — "The  areas  of  two  similar  \  po'^iwons  (  ^^^ 
proportional  to any  two  homologous  sects." 

632. — Devise  a  method  of  constructing  a  polygon  similar  to  a 


AKHAL  RELATIONS.  89 

given  polygon  and  bearing  a  given  areal  ratio  to  it, — 

633.— similar  to  one  of  two  given  dissimilar  polygons  and 
equivalent  to  the  other. 

PROBLEMS. 


634. — Devise  a  method  for  transforming  a  given  trigon  into  an 
equivalent  trigon  one  angle  of  which  shall  equal  a  designated 
angle  of  the  given  trigon,  and  one  side  of  which  shall  be  adja- 
cent to  prescribed  angle  and  shall  equal  a  given  sect; — 

635. — Which  shall  have  a  given  angle  and  a  side  of  given 
length  ;— 

636. — Which  shall  have  two  sides  of  given  lengths. 

637.— Devise  a  method  of  drawing  from  a  given  point  in  one 
side  of  a  trigon  a  line  such  that  it  shall  divide  the  trigon  into 
two  equivalent  parts ; — 

638. — Two  lines  such  that  they  shall  divide  the  trigon  into 
three  equivalent  parts ;— three  lines  such  that  they  shall  divide 
it  into  four  equivalent  parts,  etc. ; — 

639. — A  line  such  that  it  shall  divide  the  trigon  into  two  parts 
whose  areas  are  proportional  to  two  given  sects. 

640. — Find  a  point  within  a  trigon  such  that  if  lines  be  drawn 
from  it  to  the  vertices  of  the  trigon,  the  trigon  will  be  divided 
into  three  equivalent  parts ;— three  parts  whose  areas  shall  be 
proportional  to  the  sides  of  the  original  trigon  pertaining  to  them. 

641. — Transform  a  given  parallelogram  into  an  equivalent  par- 
allelogram,—(i),  which  shall  have  one  of  its  angles  equal  to  a 
given  angle;  (ii),  which  shall  have  one  of  its  angles  equal  to  a 
given  angle  and  each  leg  equal  to  a  given  sect ;  (iii),  which  shall 
have  its  leg  equal  to  a  given  sect  and  its  base  equal  to  another 
given  sect. 

642— Transform  the  sum  of  two  trigons  into  an  equivalent 
square. 

643. — Find   a  square  equivalent  to  the  sum  of  two    given 

*    See  Art.  190,  page  81, 


90  AREAL  RELATIONS. 

unequal  squares, — equivalent  to  their  difference.  See  q.  no. 
621,  page  84. 

644.— Find  a  square  wfiosc  area  shall  be  to  that  of  a  given 
square  as  one  of  two  given  sects  is  to  the  other. 

645. — Find  a  trigon  which  shall  be  similar  to  two  given  simi- 
lar trigons  and  equivalent  to  their  sum, — to  their  difference. 

646. — Transform  one  of  two  given  trigons  into  an  equivalent 
trigon  which  shall  be  similar  to  the  other. 


BOOK  IL 


CURVILINEAR   FIGURES  IN   A  SINGLE 
PLANE. 


CHAPTER  I. 
CURVES  IN  GENERAL. 

196. — Curves  have  already  been  defined  (see  art.  79, 
page  26,)  and  frequent  use  has  been  made  of  one  espe- 
cial sort.  In  this  chapter  a  more  systematic  examination 
will  be  undertaken. 

197. — Curves  are  divided  into  two  principal  groups; 
graphic  curves,  and  mathematical,  or  geometric, 

curves.  All  those  curves  the  relations  between  whose 
various  points  can  not  be  reduced  to  any  mathematical 
law,  or  exact  statement,  are  called  graphic  curves :  all 
others  are  mathematical,  or  geometric,  curves.  Such 
curves  as  those  drawn  freehand  are  almost  always 
graphic.  When  we  have  to  treat  graphic  curves  math- 
ematically we  are  obliged  to  content  ourselves  with 
approximate  results,  using  mathematical  curves  as  near- 
ly like  the  given  curves  as  we  can  conveniently  get. 


92  CURVES  IN  GENERAL. 

198. — As  has  before  been  said,  the  sect  joining  any 
two  points  of  a  curve  is  called  a  chord,  and  any  open 
curve  is  called  an  arc.  Sundry  other  terms  are  also 
used  in  the  discussion  of  curves,  and  some  of  them  are 
below  defined. 

199. — The  indefinite  straight  line  through  any  two 
separate  points  of  a  curve  is  called  a  secant  [i.  e.,  cut- 
ting) line,  or,  more  briefly,  a  secant.  The  points  of  the 
curve  through  which  the  secant  passes  are  points  of 
secancy.     (See  also  art.  201,  page  92.)  ' 

200. — If  one  of  the  two  points  of  secancy  be  fixed 
while  the  other  is  made  to  move  along  the  curve,  grad- 
ually approaching  the  first  and  carrying  the  secant  with 
it,  the  secant  will  approach  more  and  more  closely  to  a 
certain  definite  position,  depending  upon  the  curve  and 
the  fixed  point  of  secancy,  as  the  movable  point  of 
secancy  comes  more  and  more  nearly  into  coincidence 
with  the  fixed  point.  The  line  occupying  this  definite 
position  in  which  the  hitherto  secant  line  is  left  as  the 
movable  point  of  secancy  becomes  consecutive  (see  art. 
43,  page  10,)  with  the  fixed  point  is  said  to  be  tangent 
to  the  curve  at  this  (fixed)  point,  and  is  called  a  tangent 
{i.  e.,  touching)  line,  or,  more  briefly,  a  tangent;  con- 
versely the  curve  is  said  to  be  tangent  to  the  line  when 
the  line  is  tangent  to  it.  The  point  at  which  a  line  is 
tangent  to  a  curve  is  called  a  point  of  tangency,  or 
point  of  contact,  or  point  of  touch. 

201. — Any    line   having   any   other   point   than   the 


CURVES  IN  GENERAL.  93 

extremity  of  a  curve  in  common  with  the  curve,  and 

not  tangent  at  that  point,  is  said  to  be  secant  at  that 

point,  and  to  intersect  (or  to  cut)  the  curve  at  that  point. 

Obviously  the  same  line  may  be  tangent  at  one  point 

and  secant  at  another  with  certain  curves ;  or  the  same 

line  may  be  tangent  at  some  points  and  secant  at  others. 

Q.  and  E.  647. — Draw  any  graphic  arc,  and  show  its  chord,  a 
secant,  and  a  tangent. 

648. — Draw  a  graphic  arc  such  that  the  same  line  may  be  tan- 
gent at  one  point  and  secant  at  another: 

649.— such  that  the  same  line  may  be  tangent  at  two  different 
points  and  secant  at  three  others: 

650.— such  that  the  same  line  may  be  both  tangent  and  secant 
at  the  same  point.  In  the  curve  obtained,  how  many  times 
would  the  tracing  point  pass  through  the  point  of  secancy  and 
tangency  in  tracing  the  curve? 

651. — Such  that  there  may  be  two  different  tangents  at  the 
same  point. 

202. — Two  curves  are  said  to  be  tangent  to  each 

other  at  any  common  point  when  they  both  have  the 
same  tangent  at  that  point.  They  are  said  to  be  tan- 
gent externally  at  any  common  point  when  they  lie 
on  opposite  sides  of  the  common  tangent  at  that  point: 
when  both  lie  on  the  same  side  of  the  common  tangent, 
the  one  not  lying  between  the  other  and  the  tangent  is 
said  to  be  tangent  internally  to  that  other.  In  both 
cases  it  is  to  be  understood  that  only  those  portions 
extending  a  very  small  distance  from  the  point  of  tan- 
gency are  to  be  considered. 

203. — Two  curves  are  said  to  intersect  (or  cut) 


94    .  CURVES  IN  GENERAL. 

each  other  at  any  common  point  when  each  has  parts 
lying  on  opposite  sides  of  the  other  at  the  common  point. 

204. — By  the  angle  between  two  curves  at  a  com- 
mon point  is  meant  the  angle  between  their  tangents  at 
the  common  point. 

205. — In  mathematical  curves,  there  are  exact  meth- 
ods of  drawing  tangents  at  given  points  on  the  curves, 
and  also  from  given  points  outside  the  curves.  With 
graphic  curves,  however,  the  draughtsman  is  obliged  to 
content  himself  with  approximate  methods  of  performing 
these  things.  When  the  tangent  is  to  be  drawn  from 
any  point  outside  the  curve,  the  ruler  is  made  to  revolve 
about  that  point  until,  in  the  draughtsman's  judgment, 
the  line  drawn  will  just  be  tangent.  This  method  is 
fairly  satisfactory,  especially  when  the  given  point  is  at 
a  considerable  distance  from  the  point  of  tangency.     In 


the  case  of  a  tangent  so  drawn,  the  point  of  tangency, 
if  required,  may  be  approximately  found  by  this  meth- 
od:— Sundry  chords  are  drawn  parallel  to  the  tangent 
and  at  as  small  intervals  as  are  practicable;  and  through 


CURVES  IN  GENERAL.  95 

their  extremities  lines  are  drawn  perpendicular  to  the 
tangent.  On  these  lines  points  are  taken  at  distances 
from  the  tangent  equal  to  the  lengths  of  the  correspond- 
ing chords,  the  points  on  the  perpendiculars  on  opposite 
sides  of  the  point  of  tangency  being  taken  on  opposite 
sides  of  the  tangent.  Through  the  points  so  obtained  a 
smooth  curve  is  to  be  drawn.  Where  the  curve  so 
obtained  cuts  the  tangent  is  approximately  the  point  of 
tangency.  Should  the  curve  so  obtained  cut  the  tan- 
gent too  obliquely,  twice  the  length  of  each  chord  may 
be  laid  off  on  the  corresponding  perpendicular,  or  thrice, 
etc. 

206. — When  the  point  of  tangency  is  given  and  we 
are  required  to  construct  the  tangent  we  may  proceed 
thus: — Draw  through  the  given  point  various  secants 
at  slight  angular  intervals,  and  so  distributed  that  some 
of  the  second  points  of  secancy  shall  be  on  one  side  of 


the  given  point  and  some  on  the  other.  About  the 
given  point  as  center  with  any  convenient  radius  strike 
a  circular  arc,  crossing  such  region  as  it  is  judged  the 
tangent  will  pass  through.    On  each  secant  at  a  distance 


g6  CURVES  IN  GENERAL. 

from  the  second  point  of  secancy  equal  to  the  radius 
used,  take  a  point,  and  through  the  points  so  obtained 
draw  a  smooth  curve.  Where  this  curve  crosses  the 
circular  arc  before  mentioned  is  a  second  point  on  an 
approximate  tangent. 

These  are  called  ''graphic  methods"  of  construction,* 
and  give  tangents  exactly  or  approximately  with  all 
curves.  The  exact,  or  "geometric,''  methods  will  be 
discussed  in  connection  with  the  curves  to  which  they 
pertain. 

Q.  and  E.  652. — Draw  any  graphic  curve,  draw  a  tangent  from 
some  point  outside,  and  find  the  point  of  tangency. 

653. — Draw  any  graphic  curve,  choose  a  point  of  tangency,  and 
draw  the  tangent,  using  the  method  discussed  in  art.  20$.  How 
may  this  method  be  modified  so  that  the  auxiUary  circle  and  the 
other  auxiliary  curve  will  intersect  less  obliquely? 

207. — A  \  ^  /^^^  I   is  said  to  be  inscribed    in   a 

(    cham    ) 

curve  when  all  its  vertices  are  points  of  the  curve.  It 
is  said  to  be  circumscribed  about  a  curve  when  all 

{sides  ) 
..   .     [•  are  tangent  to  the  curve.     When  a  recti- 
linear figure  is  inscribed  in  any  curve,  the  curve  is  cir- 
cumscribed about  the  rectilinear  figure,  and  conversely. 

208. — As  has  before  been  said,  the  unit  of  length  is 
the  length  of  a  certain  sect  chosen  arbitrarily.  We  have 
no  direct  means,  then,  of  finding  the  length  of  a  curve, 
for  we  can  not  divide  the  curve  into  parts  each  of  which 


*  See  "Traite  de  Geometrie  Descriptive"— De  la  Gournerie;   premiere  partie, 
arts.  100  and  loi. 


CURVES  IN  GENERAL.  97 

shall  be  equal  to  a  sub-multiple  of  our  unit  sect,  since 
no  appreciable  part  of  a  curve  is  straight.  We  are  thus 
reduced  to  an  indirect  method.  We  find  the  limit  (see 
art.  12,  page  3,)  of  the  length  of  an  inscribed  chain  (or 
of  the  perimeter  of  an  inscribed  polygon)  as  the  links 
(or  sides)  are  taken  shorter  and  shorter,  the  number  of 
them  being,  of  course,  correspondingly  increased,  and 
this  limit  we  take  to  be  the  length  of  the  curve;  for  as 
the  links  (or  sides)  become  shorter  and  shorter  the 
inscribed  chain  (or  polygon)  approaches  closer  and 
closer  to  coincidence  both  in  position  and  extent  with 
the  curve,  and  becomes  indistinguishable  from  it  (except 
in  thought)  when  the  links  (or  sides)  become  inappre- 
ciable. 

209. — The  lengths  of  mathematical  curves  are  capable 
of  computation;  the  lengths  of  graphic  curves  must  be 
found  graphically.  In  practical  drawing  we  usually 
content  ourselves  with  the  approximation  got  by  taking 
the  length  of  an  inscribed  chain  each  of  whose  links  is 
about  a  tenth  of  an  inch  in  length.  When  the  curve 
leaves  the  tangent  very  rapidly  as  in  the  case  of  a  circle 
whose  radius  is  an  eighth  of  an  inch  or  less,  it  is  neces- 
sary of  course  to  use  a  much  shorter  link;  but  it  is  found 
in  practice  that  with  ordinary  curves  unavoidable  errors 
in  the  work  more  than  counter-balance  the  additional 
accuracy  which  might  be  expected  from  using  a  link 
shorter  than  a  tenth  of  an  inch.  It  will  be  noticed  that 
as  the  curve  is  more  and  more  nearly  straight,  the  links 


98  CURVES  IN  GENERAL. 

of  the  auxiliary  chain  used  may  be  taken  longer  and 
longer,  without  introducing  any  considerable  error. 

210. — Finding  the  length  of  a  curve  is  called  *'recti= 

lying**  it,  for  it  is  essentially  the  same  as  finding  an 

equivalent  **right"  line,  or  sect.     For  rectifying  curves 

graphically,    a   pair  of  light  steel-spring  dividers,  the 

distance  between  whose  points  may  be  regulated  very 

exactly  by  means  of  a  thumb-screw,  is  very  convenient. 

Such  dividers  are  called  "stepping-dividers.'* 

Q.  and  E.  654. — Draw  an  arc  which  shall  have  a  chord  of  at 
least  six  inches  and  a  point  at  least  two  inches  away  from  the 
chord.  Find  the  length  of  an  inscribed  chain  all  of  whose  links 
but  the  last  shall  be  half  an  inch  long,  the  last  being  not  more 
than  half  an  inch  in  length ;  do  the  same  with  links  a  quarter  of 
an  inch  long, — an  eighth  of  an  inch, — a  sixteenth  of  an  inch. 
What  relation  exists  among  the  various  lengths  obtained.^ 

211. — The  area  of  the  plane  surface  bounded  by  any 

closed  plane  curve  is  found  by  taking  the  limit  of  the 

area  of  any  inscribed  polygon  as  its  sides  are  made 

shorter   and   shorter   indefinitely.     By   the   expression 

"the  area  of  a  closed  plane  curve"  will  be  meant  the 

area  of  the  plane  surface  bounded  by  it. 

212. — When  the  area  of  any  closed  plane  curve  is  to 
be  obtained  graphically,  by  means  of  the  methods  of 
elementary  geometry,  especially  when  it  need  be  found 
only  with  rough  approximation,  we  frequently  get  a 
polygon  whose  size  is  about  that  of  the  given  figure  by 
taking  the  vertices  at  such  points  outside  the  curve  as 
will  cause  the  parts  cut  off  by  the  sides  of  the  polygon 


CURVES  IN  GENERAL.  99 

to  balance  the  parts  added,  as  nearly  as  can  be  estimated 
by  the  eye.  A  fine  black  silk  thread,  and  a  collection 
of  fine  cambric  needles  to  insert  at  the  vertices  of  the 
polygon,  will  be  found  useful  in  this  connection,  the 
thread  being  stretched  around  the  inserted  needles  to 
represent  the  polygon.  Instruments  called  planimeters 
{i.  ^.,  plane  measurers)  are  frequently  used  in  engineers' 
offices  for  finding  the  areas  of  plane  figures,  but  they 
are  too  complicated  to  be  discussed  here. 

213. — Two  curves  are  similar  when  for  every 
polygon  or  chain  which  may  be  inscribed  in  one  a  simi- 
lar polygon  or  chain  may  be  inscribed  in  the  other,  and 
their  ratio  of  similitude  is  the  ratio  of  similitude  of  any 
two  similar  inscribed  polygons  or  chains.  Homologous 
points,  sects,  etc.,  are  determined  in  the  same  way  as 
in  the  case  of  similar  rectilinear  figures. 

Having  given  a  brief  discussion  of  some  of  the  more 
important  properties  pertaining  to  curves  in  general,  the 
simplest  geometric  curve,  the  circle,  will  now  be  more 
fully  considered. 


CHAPTER  11. 
THE  CIRCLE. 
214. — The  simplest  and  the  most  important  of  all 
geometric  curves  is  the  circle,  which  has  already  been 
defined.  (See  art.  82,  page  27.)  The  word  circle 
is  applied  indiscriminately  to  the  curve  and  to  the 
plane  surface  bounded  by  it;  but  the  context  will  denote 
with  sufficient  clearness  which  of  these  two  meanings 
is  intended.  When  it  is  to  be  especially  indicated  that 
the  curve  alone  is  denoted,  it  is  called  a  circum= 
ference. 

215. — In  connection  with  the  circle,  we  speak  of 
center,  radius,  and  diameter,  all  of  which  have  been 
defined  (arts.  83  to  87.)  Besides  these  and  arcs,  tan- 
gents, secants,  and  chords,  whose  general  meanings 
have  been  shown,  we  speak  also  of  the  things  below 
defined. 

216. — A  semi=circle  (i.  e.,  half-circle)  is  an  arc 
which  is  half  of  the  whole  circle  or  it  is  the  plane  surface 
bounded  by  such  an  arc  and  its 'chord.  When  the 
former  meaning  is  to  be  especially  denoted,  the  arc  is 
called  a  semi=circumference. 

217. — Half  of  a  semi-circle  is  a  quadrant. 


THE  CIRCLE.  '  loi 

2 1 8. — Any  circular  arc  larger  than  a  semi-circumfer- 
ence is  called  a  major  arc;  one  smaller  than  a  semi- 
circumference  is  called  a  minor  arc.  When  an  arc  is 
named  by  means  of  its  two  extremities,  the  minor  arc 
is  to  be  understood  unless  the  contrary  is  stated  or 
clearly  indicated. 

219. — Any  portion  of  a  circle  bounded  by  an  arc  of 
that  circle  and  the  chord  of  the  arc  is  called  a  segment 
of  that  circle,  major  if  the  arc  is  major,  minor  if  the 
arc  is  minor.  The  chord  is  the  base  of  the  segment,  its 
ends  the  two  '^ertices;  the  distance  of  the  farthest 
point  of  the  arc  from  the  chord  is  the  altitude  of  the 
segment.  Such  a  segment  as  has  just  been  described 
is  a  segment  of  one  base.  The  portion  of  a  circle 
bounded  by  two  parallel  chords  of  the  circle  and  the 
arcs  intercepted  by  them  is  called  a  segment  of  two 
bases.  The  ends  of  the  bases  are  the  vertices  of  the 
segment.  The  distance  between  the  bases  is  the  alti- 
tude. Unless  the  contrary  is  stated  or  implied,  a  seg- 
ment is  to  be  understood  to  have  but  one  base. 

220. — Any  portion  of  a  circle  bounded  by  an  arc  of 
that  circle  and  the  radii  drawn  to  its  extremities  is  called 
a  sector  of  the  circle,  or  a  circular  sector,  or  more  brief- 
ly and  usually,  a  sector.  It  is  major  if  its  arc  is  major, 
minor  if  its  arc  is  minor.  The  extremities  of  the  arc 
and  the  center  of  it  (see  art.  83,  page  2'],  for  meaning  of 
* 'center  of  an  arc")  are  the  vertices  of  the  sector. 
When  we  speak  of  tJie  vertex  of  a  sector  tJie  one  al  the 


I02  THE  CIRCLE. 

ce7iter  of  the  arc  is  meant.     The  angle  of  the  sector  is 

the  angle  at  the  vertex. 

221. — By  the  centric  angle  of  any  arc  is  meant  the 
one  at  the  vertex  of  the  sector  of  which  the  arc  forms 
the  curvilinear  boundary.  The  centric  angle  of  any  arc 
is  said  to  intercept  the  arc,  and  to  be  subtended  {i.  e., 
stretched  across  or  under)  by  the  arc,  also  by  the  chord 
of  the  arc.  We  say  also  that  an  arc  is  subtended  by  its 
chord. 

222. — An  arc  is  produced  by  being  extended  so  that 
the  portion  thus  obtained  shall  be  an  are  about  the  same 
center  as  the  first.  Any  arc  is  added  to  another  of 
equal  radius  by  producing  that  other  until  the  produced 
part  equals  the  first.  The  resulting  arc  is  the  sum  of 
the  two  given  arcs. 

{    quadrant    ) 
223. — Two  arcs  whose  sum  is  a  ■<  semi-circle  V  are 

(       circle       ) 
i   complementary   )  r   complement  \ 

<   supplementary    >  and  each  is  the  ■<    supplement    >- 
(    explementary    )  (    explement    ) 

of  the  other. 

224. — Circumferences  are  divided  into  360  equal 
parts,  each  of  which  is  called  a  degree  of  arc  or  an  arc 
degree.  Each  degree  of  arc  is  divided  into  60  equal 
parts  each  of  which  is  called  a  minute  of  arc;  each 
minute  of  arc,  into  60  equal  parts  each  of  which  is  called 
a  second  of  arc.     The  symbols  for  degrees,  minutes, 


THE  CIRCLE.  103 

and  seconds  of  arc  are  the  same  as  in  the  case  of  angles. 
Evidently  the  size  of  a  degree  of  arc  depends  upon  the 
size  of  the  radius  of  the  arc. 

225. — If  the  vertex  of  an  angle  is  some  point  (except 
an  extremity)  of  the  arc  of  a  segment,  and  its  sides  pass 
through  the  vertices  of  the  segment,  the  convex  angle 
between  the  two  sides  is  said  to  be  inscribed  in  the 
segment.  An  inscribed  angle  is  said  to  intercept  the 
arc  included  between  its  sides  and  explementary  to  the 
arc  of  the  segment  in  which  the  angle  is  inscribed. 

226. — Any  convex  angle  is  inscribed  in  a  circle  if 

its  vertex  is  a  point  of  the  circumference  and  its  sides 
are  chords  or  chords  produced.  The  centric  angle  inter- 
cepting the  same  arc  as  an  inscribed  angle  is  said  to  be 
the  corresponding  centric  angle. 

Q.  and  E.  655. — Is  the  circle  sym-centric  ? — sym-axic?  If  so, 
how  many  sym-centers  has  any  one  circle? — sym-axes? 

656.— What  relation  exists  among  the  various  sym-axes  of  a 
circle,  if  it  has  more  than  one.? 

657.— What  relation  exists  between  a  sym-axis  and  a  diameter? 

658.— Same  three  questions  for  segments  of  one  base, — of  two 
bases. 

659.— Same  for  sectors. 

65o. — How  many  radii  has  a  circle? 

661. — What  relation  exists  among  them?  Give  your  reason 
for  your  answer. 

662. — What  relation  exists  between  a  diameter  and  a  radius  in 
the  same  circle? 

663.— What  is  the  longest  chord  that  can  be  drawn  in  a  circle? 

664. — What  relation  exists  between  the  two  arcs  into  which  it 
divides  the  circuiuierence? — the  two  segments  into   which   it 


I04  THE  CIRCLE. 

divides  the  circle? 

665.— How  may  the  shortest  line  from  any  given  point  to  a 
point  of  a  given  circle  be  drawn?— the  longest  line? 

666.— What  relation  exists  between  two  circles  having  equal 
radii  ? — equal  diameters  ? 

667.— Draw  six  circles  of  different  sizes  and  obtain  the  approx- 
imate lengths  of  their  circumferences  by  the  method  discuss'ed  in 
art.  208,  page  96.  What  relation,  if  any,  exists  among  the  ratios 
these  lengths  bear  to  the  lengths  of  the  corresponding  radii? 
Take  no  circle  with  a  radius  of  less  than  two  inches. 

668.— At  what  point  of  a  chord  is  it  met  by  a  perpendicular 
dropped  from  the  center  of  the  circle? 

669. — How  might  this  have  been  known  from  your  conclusion 
in  the  case  of  q.  no.  88? 

670.— If  two  chords  of  the  same  circle  or  of  equal  circles  are 
equi-distant  from  the  center,  or  centers  of  the  corresponding  cir- 
cles, what  relation  exists  between  their  lengths? 

671. — What,  if  they  are  unequally  distant  from  the  center,  or 
from  the  centers  of  their  respective  circles? 

672. — What  relation  exists  between  the  distances  from  the 
center,  of  two  equal  chords  in  the  same  circle? — two  unequal 
chords?  Show  how  the  conclusions  which  you  come  to  in  this 
case  might  have  been  known  by  means  of  those  arrived  at  in 
response  to  the  two  preceding  questions. 

673.— What  is  the  locus  of  the  middle  point  of  a  chord  parallel 
to  a  given  line? 

674.— What  relation  exists  between  the  arcs  intercepted  by 
two  parallel  chords? 

675. — If  a  diameter  be  perpendicular  to  a  chord  what  relation 
exists  between  the  two  parts  into  which  it  divides  each  arc  sub- 
tended by  the  chord?— the  centric  angle  of  the  arc  subtended  by 
the  chord? 

676.— If  two  arcs  are  equal  what  relation,  if  any,  exists 
between  their  chords? 

677.— What,  if  the  arcs  be  unequal  ?    Always  ? 

678.— If  the  chords  of  two  arcs  are  equal,  what  relation,  if  any, 
exists  between  the  arcs?    Always? 


THE  CIRCLE.  105 

679. — What,  if  the  chords  be  unequal  ?  How  might  this  be 
known  through  your  conclusion  in  reply  to  q.  676? 

680. — Supply  the  missing  words  in  the  following  proposition: 
—"If  arcs  of radii  have  equal  chords  the  arcs  are 

681.— In  the  case  of  unequal  arcs,  are  the  chords  proportional 
to  the  arcs? 

682.— If  so,  show  why ;  if  not,  show  why  not.  If  in  some 
cases  they  are  and  in  other  cases  they  are  not,  show  what  other 
relation  exists  between  the  arcs  when  they  are,  which  does  not 
exist  between  them  when  they  are  not. 

682. — If  two  arcs  are  equal,  what  relation,  if  any,  exists 
between  their  centric  angles? — what,  if  the  arcs  are  unequal  ? 

683. — What  relation  exists  between  two  arcs  subtending  equal 
centric  angles?    Always? 

684.— What,  if  the  centric  angles  are  unequal  ?  How  might 
this  have  been  known  from  the  conclusion  to  q.  682? 

685.— What  relation  exists  between  the  centric  angle  of  the 
sum  of  two  arcs  (of  equal  radii)  and  the  sum  of  their  respective 
centric  angles?— the  centric  angle  of  the  difference  between  two 
arcs  (of  equal  radii)  and  the  difference  between  their  respective 
centric  angles? 

686.— Supply  the  missing  words  in  the  following  proposition: 

—"Arcs  of are to  their  centric  angles ; 

and,  conversely,  centric  angles  are to  the  arcs  sub- 
tending them  if  the  arcs  have " 

687.— What  relation  exists  between  the  number  of  degrees, 
minutes,  and  seconds  in  any  arc  and  the  number  of  degrees, 
minutes,  and  seconds  in  its  centric  angle? 

(  complementary  ) 
688. — If  two  arcs  are  \  supplementary  [•  what  relation  exists 
(  explementary  ) 
between  their  respective  centric  angles? 

689.— What  relation  does  the  perpendicular  to  a  radius  through 
its  outer  extremity  bear  to  the  circle? 

690.— What  relation  does  any  oblique  line  through  the  outer 
extremity  of  a  radius  bear  to  the  circle? 


io6  THE  CIRCLE. 

691. — How  many  tangents  may  a  circle  have  at  any  one  point 
of  tangency? 

692.— How  many  tangents  can  be  drawn  to  a  circle  from  any 
point  outside  a  circle? — from  any  point  inside?  Give  reasons 
for  your  answers  to  the  last  two  questions. 

693. — What  angle  does  a  radius  drawn  to  a  point  of  tangency 
make  with  the  tangent? 

694. — If  the  various  tangents  be  drawn  from  a  given  point  out- 
side a  circle,  what  relation,  if  any,  exists  among  the  distances  of 
the  points  of  tangency  from  the  given  point? 

655. — What  relation  exists  between  the  sums  of  sets  of  alter- 
nate sides  in  any  circumscribed  polygon  of  an  even  number  of 
sides? 

696. — What  relation  do  the  tangents  from  any  one  point  bear 
to  the  line  through  the  given  point  and  the  center  of  the  circle? 

697. — What  relation  do  they  bear  to  the  chord  (or  chords)  join- 
ing the  points  of  tangency  ? 

698.— What  kind  of  line  in  the  circle  is  that  which  joins  the 
points  of  tangency  of  two  parallel  tangents  to  the  same  circle? 

699. — What  relation  exists  between  the  arcs  intercepted  by 
two  such  tangents?  Show  the  connection  between  this  relation 
and  that  developed  in  reply  to  the  preceding  question. 

700.— What  relation  exists  between  the  arcs  intercepted 
between  a  tangent  and  a  chord  parallel  to  it? 

701. — In  each  of  three  or  four  circles  of  different  sizes  draw 
three  or  four  inscribed  angles  of  various  sizes,  and  compare  each 
with  its  corresponding  centric  angle.  What  relation,  if  any, 
seems  to  exist  between  each  inscribed  angle  and  its  correspond- 
ing centric  angle? 

702. — What  relation,  if  any,  exists  among  all  the  angles  which 
it  is  possible  to  inscribe  in  a  given  segment? 

703. — What  relation,  if  any,  exists  between  the  centric  angle 
subtended  by  a  chord,  and  the  acute  angle  between  that  chord 
and  a  tangent  through  one  end  of  it? 

704.— If  two  chords  AB  and  CF  intersect  at  a  point  P,  what 
relation  does  the  angle  APC  bear  to  the  centric  angle  intercepting 
the  sum  of  the  arcs  AC  and  BF?    Trace  the  connection  between 


THE  CIRCLE.  107 

this  relation  and  that  developed  by  the  preceding  two  questions 
in  connection  with  q.  701. 

705. — What  relation  does  the  angle  between  two  secants 
drawn  from  a  point  without  a  circle  bear  to  the  angle  at  the  cen- 
ter intercepting  the  different*  between  the  arcs  intercepted  by 
the  secants?  Trace  the  connection  between  the  relation  here 
developed  and  those  developed  just  before  it. 

706. — Same  for  the  angle  between  two  tangents  drawn  from 
the  same  point. 

707.— Same  for  the  angle  between  a  tangent  and  a  secant 
drawn  from  the  same  point. 

708. — What  relation  exists  between  the  alternate  angles  of  an 
inscribed  tetragon.?  * 

709. — Are  two  unequal  circles  similar  figures?— homothetic?  If 
homothetic  are  they  directly  homothetic  or  inversely  homothetic? 

710.— What  relation  exists  between  the  ratio  of  their  circum- 
ferences and  that  of  the  corresponding  radii  ? 

711.— What  is  the  locus  of  a  point  half-way  from  a  given  point 

111 
to  a  point  upon  a  given  circle.? — ths  of  the  way? 

712.— Is  this  locus  sym-axic?— if  so,  about  what  axis? 
713. — Is  it  sym-centric?— if  so,  about  what  point? 
714. — Does  its  size  depend  upon  the  position  of  the  given 
point? — upon  the  size  of  the  given  circle? — upon  the  ratio  — ? 
715 —Does  its  position  depend  upon  the  position  of  the  given 


m 


point?— upon  the  position  of  the  given  circle? — upon  the  ratio 

716.— Do  all  of  your  conclusions  in  connection  with  this  locus 
hold  true  when  m  >  nt 

7ij.—\f  two  chords  AB  and  CF  in  any  circle  intersect  at  the 
point  P,  what  relation  exists  between  the  trigons  APC  and  BPF? 

AP  PF 

—the  ratios  g^  and  pg?— the  products  APxPB  and  CPxPF? 

718. — If  two  secants  are  drawn  from  any  point  P  without  a  cir- 


*  The  relations  which  the  student  will  develop  in  investigating  the  cases  pro- 
posed in  q.  no.  701--707  are  very  useful  in  comparing  the  different  angles  of 
inscribed  figures. 


io8  THE  CIRCLE. 

cle,  one  cutting  the  circle  in  the  two  points  A  and  B,  and  the 
other  in  the  two  points  C  and  F,  A  and  C  being  nearer  P  than  B 
and  F  respectively,  what  relation  exists  between  the  trigoiis  ARC 

AP  PF 

and  BPF? — between  the  ratios  ^-p  and  prj  ? — between  the  pro- 
ducts APxPB  and  CPxPF? 

719.— Do  these  relations  still  hold  when  the  secant  PAB  has 
revolved  about  P  until  it  has  become  tangent,  so  that  the  points 
A  and  B  merge  into  one? 

720.— What  kinds  of  trigons  may  be  inscribed  in  circles? — cir- 
cumscribed about  them  ? 

721. — Same  for  tetragons.    Give  reasons  for  your  answers. 

722. — What  is  the  locus  of  the  vertex  of  a  trigon  whose  base  is 
fixed  and  whose  angle  at  the  vertex  is  constant? 

723. — Develop  a  formula  for  the  area  of  a  circumscribed  poly- 
gon in  terms  of  its  perimeter  and  the  radius  of  the  inscribed  cir- 
cle.   See  q.  610,  page  81. 

724 — Develop  a  formula  for  the  area  of  a  circle  in  terms  of  its 
radius  and  its  circumference.  * 

725.  —Develop  a  formula  for  the  area  of  a  sector  in  terms  of  its 
radius  and  its  arc. 

726. — What  relation  exists  between  the  areal  ratio  of  similar 
sectors,  similar  segments,  similar  circles,  etc.,  and  their  ratio  of 
similitude? 

727. — What  relation  exists  between  the  areal  ratio  of  any  two 
similar  plane  figures  and  their  ratio  of  similitude?    Show  why. 

PROBLEMS. 


728. — Find  the  locus  of  one  end  of  a  given  sect  which  has  the 
other  end  on  a  given  circle  and  is  parallel  to  a  given  line. 

729. — Find  the  locus  of  a  point  on  a  tangent  to  a  given  circle 
and  at  a  given  distance  from  the  point  of  tangency. 

730. — Find  the  locus  of  a  point  at  a  given  distance  from  a  giv- 
en circle. 


This  formula  being  correctly  developed  a  discussion  of  the  ratio  of  the  cir- 
cumference to  the  diameter  may  be  introduced. 


THE  CIRCLE.  109 

731.— Draw  a  sect  of  given  length  which  shall  be  parallel  to  a 
given  line  and  have  its  extremities  in  two  given  circles. 

Draw  the  circle  which  shall 

732.— Pass  through  three  given  points,— (i),  three  points  inde- 
pendent; (ii),  collinear; 

733. — Pass  through  four  given  points; 

734.— Pass  through  two  given  points  and  have  a  given  radius ; 

735. — Pass  through  a  given  point,  have  a  given  radius,  and  be 
tangent  to  a  given  line ; 

736. — Have  a  given  radius,  and  be  tangent  to  two  given  lines; 

737. — Be  tangent  to  three  given  lines,— (i),  lines  all  parallel ; 
fii),  two  lines  only,  parallel ;  (iii),  no  two  lines  parallel ;  (iv), 
lines  concurrent. 

738. — Pass  through  two  given  points  and  be  tangent  to  given 
line,— (i),  when  sect  joining  given  points  is  parallel  to  given  line; 
(ii),  when  it  is  not  parallel ;  (iii),  one  of  given  points  is  the  point 
of  tangency.    See  q.  718,  page  107. 

739.— Pass  through  a  given  point  and  be  tangent  to  two  given 
lines, — (i),  lines  parallel;  (ii),  lines  not  parallel;  (iii),  point  or 
one  of  given  lines;  (iv),  point  equidistant  from  two  given  lines. 
In  (i)  and  (ii)  make  use  of  fact  that  the  locus  of  a  point  equidis 
tant  from  the  two  given  lines  is  a  sym-axis,  and  compare  with 
preceding  problem. 

740. — Pass  through  a  given  point,  be  tangent  to  a  given  line 
and  have  a  given  radius— (i), given  point  on  given  line;— (ii),  not. 

741. — Be  tangent  to  given  circle  and  to  given  line,  and  have 
given  point  of  tangency;  (i),  on  given  line;  (ii),  on  given  circle. 
Two  solutions  under  each,  according  as  two  circles  are  tangent 
internally  or  externally. 

742. — Pass  through  a  given  point,  have  a  given  radius,  and  be 
tangent  to  a  given  circle,— (i)  given  point  on  given  circle;  (ii)  not. 

743. — Have  a  given  radius  and  be  tangent  to  a  given  line  and 
to  a  given  circle,— (i),  the  two  circles  tangent  internally ;  (ii), 
tangent  externally. 

744.— Have  a  given  radius  and  be  tangent  to  two  given  circles, 
— (i),  tangent  externally  to  both  given  circles;  (ii),  tangent  inter- 
nally to  one,  or  conversely,  and  externally  to  the  other.    Two 


no  THE  CIRCLE. 

cases  under  second  division.    How  many  solutions  in  all? 

745. — Draw  within  a  given  circle  three  circles  equal  to  each 
other  and  each  tangent  to  the  given  circle  and  to  the  other  two. 

746. — About  the  vertices  of  a  trigon  as  centers  describe  circles 
so  that  each  pair  shall  be  tangent  to  each  other  and  to  the  third, 
— (i),  internally ;— (ii),  externally. 

Construct  the  trigons  which  shall  have  given  values  for 

747.  ^,  b,  and  m^. 

748.  <2,  b,  and  hf,. 

749.  «,  m^^  and  ^b 

750.  «,  m^,  and  a. 

751.  a,  Wb,  and  h\,. 

752.  ^,  m\,^  and  h^. 

753.  a,  Wb?  and  a. 

754.  a,  m\i,  and  ^. 

755 .  a,  //a,  and  h\,. 

756.  a,  //a,  and  a. 

757.  a,  //b,  and  kc. 

758.  «,  >^b,  and  ^b- 

759.  ^,  ^b,  and  dc. 

760.  <7,  /^b,  and  a. 

761.  «,  /^b,  and  /5. 

762.  ^,  >^b,  and  p. 

763.  //^a,  ?'^b,  and  a. 

764.  //?a,  /^b,  and  y. 

765.  -^a,  >^b,  and  a. 

766.  >^a,  «,  and  p. 

767.  «,  <5'— <:,  and  h\,. 

768.  «,  <5'— (T,  and  kc, 
765.    /,  a-f-/'?,  and  he. 

770.  ^,  (^,  and  R.  * 

771.  <3;,  ?;^a,  and  R. 


R  is  radius  of  circumscribed  circle:  r  of  inscribed. 


THE  CIRCLE.  m 

772.  a,  m\,,  and  R, 

773.  a^  kg,,  and  R. 

774.  a,  /^b,  and  R. 
j-j^.  a,  /?,  and  R. 

776.  <a;,  /^,  and  /-. 

777.  ;;^a,  >^a,  and  /?. 

778.  Ma,  «,  and  A\ 

779.  /^a,  <^a,  and  i?. 

780.  /^a,  «,  and  -/?. 

781.  //a,  A^,  and  A\ 

782.  </a,  «,  and  r. 

783.  «,  /^,  and  r. 

784.  a,  /3,  and  A'. 

785.  a,  i^-y,  and  7?. 

786.  /^+^,  ;3,  and  i?. 

Construct  the  isosceles  trigon  which  shall  have  given  values 
for 

787.  b  and  R. 

788.  c  and  R. 

789.  <:  and  r. 

Construct  the  right  trigon  which  shall  have  given  values  for 

790.  a  and  r. 


791. 

a  and  R. 

792. 

c  and  h^. 

793- 

c  and  Ma. 

794- 

c  and  r. 

795. 

p  and  ^0. 

796. 

p  and  r. 

797- 

/  and  R. 

798. 

p  and  «— /3. 

799. 

^— r?  and  r. 

800. 

c—a  and  /?. 

801. 

^c  and  /-. 

112  THE  CIRCLE. 

802.  h^  and  R. 

803.  d^  and  r. 

804.  a  and  /■. 

805.  a  and  R. 

806.  «— /3  and  r.  ' 

807.  a-/3and7?. 

Construct  the  tetragon  which  shall  have  given  values  for 

808.  a,b,g\,  /.ad,  and  /.cd. 

Construct  the  trapezoid  which  shall  have  given  values  for 

809.  a,  b,  c,  and  d. 

810.  a,  b,  d,  and  m. 

811.  a,  b,p,  and  7n. 

812.  a,  c,  g\,  and  ^2. 
813-    ci,g-u£'2,  and  m. 

227. — A  mixtilinear  figure,  t.  c.j  one  whose  out- 
line is  made  up  of  curves  and  right  lines,  is  said  to  be 
inscribed  in  a  polygon  when  its  vertices  lie  on  the 
sides  of  the  polygon  and  its  curvilinear  sides  are  tan- 
gent to  those  sides  of  the  polygon  on  which  no  vertices 
of  the  inscribed  figure  are. 

814. — Inscribe  a  semicircle  in  a  given  trigon,  so  that  the  verti- 
ces of  the  semicircle  shall  both  rest  on  the  same  side  of  the  tri. 
gon.  How  many  solutions?  Does  the  character  of  the  trigon 
have  anything  to  do  in  determining  the  number  of  solutions?  If 
so,  what  and  why? 

815. — Same,  except  that  base  of  semicircle  shall  be  parallel  to 
a  given  line  which  is  not  parallel  to  any  side  of  the  trigon. 
Same  questions  as  in  preceding  exercise. 

816.— Inscribe  in  a  given  trigon  a  segment  similar  to  a  given 
major  segment, — (i),  base  of  segment  on  one  of  the  sides  of 
trigon ;  (ii),  base  of  segment  not  on  any  side,  but  parallel  to  a 
given  line.    Same  questions  as  in  no.  814, 

817.— Same  for  minor  segment,  and  same  questions  as  in  814. 


THE  CIRCLE.  113 

818. — Same  for  major  sector,— (i),  one  of  rectilinear  sides  on 
one  side  of  trigon ;  (ii),  not  on  any  side  but  parallel  to  a  given 
line     Same  questions  as  in  no.  814. 

819.— Same  for  minor  sector  and  same  questions  as  in  no.  814. 

820  to  825. — Same  as  from  no.  814  to  no.  819  except  that  three 
indefinite  non-concurrent  lines,  no  more  than  two  of  which  may 
be  parallel,  are  to  be  substituted  for  the  sides  of  the  given  trigon. 
Discuss  each  one  fully.  Problems  from  no.  814  to  no.  825  inclu- 
sive may  most  easily  be  solved  by  making  use  of  the  principles 
of  homothesy,  using  one  of  the  intersections  of  the  given  straight 
lines  as  center  of  homothesy. 

826. — Construct  a  circle  whose  perimeter  shall  equivale  the 
sum  of  the  perimeters  of  two  given  unequal  circles, — whose  area 
shall  equivale  the  sum  of  their  areas. 

827. — Construct  a  semicircle  whose  perimeter  shall  equivale 
the  difference  between  the  perimeters  of  two  given  unequal  semi- 
circles,—whose  area  shall  equivale  the  difference  between  their 
areas. 

828.— Construct  a  sector  which  shall  be  similar  to  a  given  sec- 
tor and  three  times  as  large. 

829. — Construct  a  segment  which  shall  be  similar  to  a  given 
segment  of  two  bases,  and  half  as  large. 

For  problems,  no.  826-7-8-9,  see  art.  191,  page  83,  and  q.  727, 
page  loS. 


The  Department  of  Mathematics,  ^ 

The  State  Agricultural  College,       [ 

Port  Collins,  Colorado.  ) 

Dear  Sir  : 

The  accompanying  copy  of  "An  Inductive  Manual  of  the  Straight 
Line  and  the  Circle  "  is  sent  you  for  examination,  with  view  to  introduc- 
tion and  use  in  your  school . 

Your  attention  is  particularly  called  to  the  following  features,  which,  i1 
is  believed,  distinguish  the  book  from  other  text  books  on  elementary  plant 
geometry  : 

I.  The  constancy  with  which  there  is  kept  before  the  mind  of  the  stu 
dent  the  connection  existing  between  geometric  relations  and  their  applica 
tions  in  the  arts,— e.  g.,  Articles  53,  65,  69,  127,  141,  142,  205,  206,  etc.,  and  th( 
sxercises  coming  under  them. 

II.  The  early  introduction  and  the  large  use  of  the  notion  of  locus. 

III.  The  considerable  development  of  the  notion  of  symmetry. 

IV.  The  distinction  between  the  obverse  and  the  reverse  of  plant 
figures  ;  see  Art.  120. 

V.  The  closeness  of  relation  between  regular  chains,  regular  polygons 
md  the  circle  ;  see  Articles  150-160. 

VI .  The  great  number  of  exercises  and  problems,  and  the  logical  ordei 
3f  their  arrangement. 

Besides  the  features  above  noted,  there  is,  of  course,  the  fundamental 
idea  of  the  entire  book, — viz.,  to  furnish  the  student  the  tools  and  tht 
material,  and  let  him  work  up  his  ideas  for  himself,  helping  him  by  as  skil 
fully  asked  questions  as  the  author  has  been  able  to  devise  whenever  there 
is  need  for  help,  but  in  all  cases  leaving  some  actual  work  and  thought  to  the 
student  himself. 

The  author  respectfully  solicits  your  opinion  of  the  book,  with  permis- 
sion to  publish  the  opinion  if  he  desires  so  to  do. 

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be  addressed  to  the  author,  at  Port  Collins,  Colorado,  or  to  Mr.  Alva  M. 

Meyers,  Litchfield,  Michigan . 

William  J.   Meyers. 


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